On the graded algebras associated with Hecke symmetries, II. The Hilbert series

TL;DR

This paper investigates the properties of graded algebras linked to Hecke symmetries, extending previous work by removing restrictions on the parameter q, and analyzing their Hilbert series.

Contribution

It advances understanding of graded algebras associated with Hecke symmetries by generalizing results to arbitrary q values without prior restrictions.

Findings

Characterization of Hilbert series for these algebras

Extension of properties to all q parameters

Deeper insight into algebraic structures related to Hecke symmetries

Abstract

This paper continues the work which attempts to understand the general properties of the graded algebras associated with Hecke symmetries without a restriction on the parameter q of the Hecke relation imposed in earlier results.

Equations

$$1+q+\dots +q^{n-1}\ne 0for\; alln>0.$$

$$H_A(t)=\sum (\dim A_n)t^n.$$

$$H_{\Lambda (V,R)}(t)=f_0(-t)/f_1(t),H_{S(V,R)}(t)=f_1(-t)/f_0(t)$$

$$\prod _{i=1}^{r_0}(1+{\alpha }_{i}t)\cdot \prod _{j=1}^{r_1}(1-{\beta }_{j}t{)}^{-1}\mathrm{and}\prod _{j=1}^{r_1}(1+{\beta }_{j}t)\cdot \prod _{i=1}^{r_0}(1-{\alpha }_{i}t{)}^{-1}$$

$$(R\otimes \mathrm{Id})(\mathrm{Id}\otimes R)(R\otimes \mathrm{Id})=(\mathrm{Id}\otimes R)(R\otimes \mathrm{Id})(\mathrm{Id}\otimes R)$$

$$(R-q\cdot \mathrm{Id})(R+\mathrm{Id})=0where0\ne q\in \text{mathchar}1404.$$

$$\openup 3.0pt\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr 0.0pt{\hfil$\displaystyle T_{i}T_{j}=T_{j}T_{i}\quad\hbox{whenever $|i-j|>1$},\cr 0.0pt{\hfil$\displaystyle T_{i}T_{i+1}T_{i}=T_{i+1}T_{i}T_{i+1}\quad\hbox{for $i=1,\ldots,n-2$},\cr 0.0pt{\hfil$\displaystyle(T_{i}-q)(T_{i}+1)=0\quad\hbox{for $i=1,\ldots,n-1$}.\cr}}}}$$

$$R_i^{(n)}={\mathrm{Id}}^{\otimes (i-1)}\otimes R\otimes {\mathrm{Id}}^{\otimes (n-i-1)}.$$

$$A(R)=\underset{n=0}{\overset{\infty }{⨁}}{A}_n(R)$$

$$A_n(R)\cong (\mathrm{Coend}V{)}^{\otimes n}/I_n$$

$${\mathrm{End}}_{{\mathcal{H}}_n}{V}^{\otimes n}=\bigcap _{i=1}^{n-1}{E}_i^{(n)}$$

$$E_i^{(n)}=({\mathrm{End}}_{\text{mathchar}1404}V{)}^{\otimes (i-1)}\otimes ({\mathrm{End}}_{{\mathcal{H}}_2}{V}^{\otimes 2})\otimes ({\mathrm{End}}_{\text{mathchar}1404}V{)}^{\otimes (n-i-1)}$$

$$I_n=\sum _{i=1}^{n-1}{E_i^{(n)}}^{\perp }=\sum _{i=1}^{n-1}(\mathrm{Coend}V{)}^{\otimes (i-1)}\otimes I_2\otimes (\mathrm{Coend}V{)}^{\otimes (n-i-1)}.$$

$$T(\mathrm{Coend}V)=\underset{n=0}{\overset{\infty }{⨁}}(\mathrm{Coend}V{)}^{\otimes n}$$

$$A_{n}(R)=\bigl{(}\mathop{\rm End}\nolimits_{{\cal H}_{n}}\mskip-5.0muV^{\otimes n}\bigr{)}^{*}.$$

$$\mathrm{Grot}(R)=\underset{n=0}{\overset{\infty }{⨁}}{\mathrm{Grot}}_n(R).$$

$$\mathrm{Im}(R-q\cdot \mathrm{Id})\subset V^{\otimes 2}\mathrm{and}\mathrm{Ker}(R-q\cdot \mathrm{Id})\subset V^{\otimes 2}.$$

$$\cdots \to \mathrm{Sym}(3)\to \mathrm{Sym}(2)\to \mathrm{Sym}(1)\to Z$$

$${\mathcal{I}}_{\lambda }=\{i\in N\mid 1\le i<|\lambda |\; andi\ne {\lambda }_1+\dots +{\lambda }_{j}for\; eachj=1,\dots ,\ell (\lambda )\}$$

$${\mathcal{P}}^2(n)=\{(\lambda ,\mu )\in \mathcal{P}\times \mathcal{P}\mid |\lambda |+|\mu |=n\}.$$

$$({\lambda }_1,\dots ,{\lambda }_{\ell (\lambda )},{\mu }_1,\dots ,{\mu }_{\ell (\mu )}).$$

$$\eqalign{{\cal I}_{\lambda,\mskip 1.0mu\mu}={\cal I}^{0}_{\lambda,\mskip 1.0mu\mu}\cup{\cal I}^{1}_{\lambda,\mskip 1.0mu\mu}\quad{\rm where}\quad{\cal I}^{0}_{\lambda,\mskip 1.0mu\mu}&{}=\{i\in{\cal I}_{\lambda,\mskip 1.0mu\mu}\mid i<|\lambda|\}={\cal I}_{\lambda}\,,\cr{\cal I}^{1}_{\lambda,\mskip 1.0mu\mu}&{}=\{i\in{\cal I}_{\lambda,\mskip 1.0mu\mu}\mid i>|\lambda|\}=\{|\lambda|+i\mid i\in{\cal I}_{\mu}\}\,.}$$

$$\chi_{\lambda,\mskip 1.0mu\mu}(T_{i})=\cases{q&for $i\in{\cal I}^{0}_{\lambda,\mskip 1.0mu\mu}\,$,\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr-1&for $i\in{\cal I}^{1}_{\lambda,\mskip 1.0mu\mu}\,$,}$$

$$[n{]}_q=1+q+\cdots +q^{n-1}.$$

$$V^{\otimes n}\cong \underset{\lambda \in \mathcal{P}(n)}{⨁}{V}_R^{\lambda }\otimes S^{\lambda }where{V}_R^{\lambda }={\mathrm{Hom}}_{{\mathcal{H}}_n}(S^{\lambda },V^{\otimes n}).$$

$${\mathrm{End}}_{{\mathcal{H}}_n}{V}^{\otimes n}\cong \prod _{\lambda \in \mathcal{P}(n)}{\mathrm{End}}_{\text{mathchar}1404}{V}_R^{\lambda }.$$

$$\{V_R^{\lambda }\mid \lambda \in \mathcal{P},V_R^{\lambda }\ne 0\}$$

$$[X]\cdot [Y]=[{\mathrm{ind}}_{m,n}^{m+n}X\otimes Y]$$

$$[S^{\mu }]\cdot [S^{\nu }]=\sum _{\lambda \in \mathcal{P}}{c}_{\mu \nu }^{\lambda }[S^{\lambda }]for\mu ,\nu \in \mathcal{P}$$

$$s_{\mu }{s}_{\nu }=\sum _{\lambda \in \mathcal{P}}{c}_{\mu \nu }^{\lambda }{s}_{\lambda }.$$

Full text

On the graded algebras associated with Hecke symmetries, II. The Hilbert series

Serge Skryabin

Institute of Mathematics and Mechanics, Kazan Federal University,Kremlevskaya St. 18, 420008 Kazan, Russia E-mail: Serge.Skryabin@kpfu.ru

Introduction

With a Hecke symmetry $R$ on a finite dimensional vector space $V$ one associates the $R$-symmetric algebra ${{S}}(V,R)$, the $R$-skewsymmetric algebra $\Lambda(V,R)$, and the bialgebra $A(R)$ given by the Faddeev-Reshetikhin-Takhtajan construction [22]. The first two algebras are noncommutative analogs of the symmetric and the exterior algebras of $V$. The algebra $A(R)$ is a noncommutative analog of the ring of polynomial functions on the space of $r\times r$ matrices. By localizing $A(R)$ one obtains a Hopf algebra which represents a nonstandard quantum group (see [15]).

For two Hecke symmetries $R,R^{\prime}$ with the same parameter $q$ of the Hecke relation there is also an algebra $A(R^{\prime},R)$ generalizing $A(R)$. This algebra introduced by Phùng Hô Hai [17] in a different notation represents a “quantum hom-space”. All these algebras are quadratic graded algebras.

The present paper continues the work started in [24] which attempts to understand the general properties of the graded algebras associated with Hecke symmetries without a restriction on the parameter $q$ of the Hecke relation imposed in earlier results. The known results rely heavily on semisimplicity of the Hecke algebras ${\cal H}_{n}={\cal H}_{n}(q)$ of type $A$ which operate in the tensor powers of the initial space $V$. This semisimple case occurs precisely when

[TABLE]

Here we will be concerned with the determination of the Hilbert series and several related results. For a graded algebra $A=A_{0}\oplus A_{1}\oplus A_{2}\oplus\ldots$ with finite dimensional homogeneous components its Hilbert series is a formal power series in one indeterminate defined as

[TABLE]

The question as to what are possible Hilbert series of the algebras ${{S}}(V,R)$ and $\Lambda(V,R)$ arose in the work of Gurevich [15]. About 10 years later Phùng Hô Hai [16] and, independently, Davydov [5] observed that in the semisimple case the dimensions of the homogeneous components of the two algebras form totally positive sequences. From this they deduced that the Hilbert series of these algebras are rational functions with negative roots and positive poles. This conclusion is based on analytic results obtained by Aissen, Schoenberg, Whitney [1] and Edrei [12] which describe the generating series of totally positive sequences.

As we have seen in [24], good properties may be lost when $q$ is a root of 1. However, it was shown there that several previously known results extend to the case of an arbitrary $q$ provided that a certain additional condition is imposed. Recall that an indecomposable ${\cal H}_{n}$-module is said to have a $1$-dimensional source if it is a direct summand of an ${\cal H}_{n}$-module induced from a $1$-dimensional representation of a parabolic subalgebra, and we say that $R$ satisfies the $1$-dimensional source condition if for each $n>0$ all indecomposable direct summands of $\,V^{\otimes n}$ regarded as an ${\cal H}_{n}$-module with respect to the representation arising from $R$ have $1$-dimensional sources. This condition is satisfied automatically in the semisimple case mentioned earlier. Our main result in the new paper is

Theorem 3.8. Suppose that $R$ satisfies the $1$-dimensional source condition. Then

[TABLE]

with integer polynomials $\,f_{0},\,f_{1}\in{{Z}}[t]\,$ whose constant terms are equal to $1$ and all roots are positive real numbers.

The pair $(r_{0},r_{1})$ where $r_{i}=\mathop{\rm deg}\nolimits f_{i}$ for $i=0,1$ is called the birank of $R$. Thus the Hilbert series of the two algebras can be written as

[TABLE]

where $\alpha_{i}$ and $\beta_{j}$ are positive algebraic integers.

The already mentioned results of Phùng Hô Hai and Davydov may be viewed as a nice application of the theory of symmetric functions. We will use a nonstandard notation $\,\mathop{\rm Sym}\nolimits\,$ for the ring of symmetric functions defined as in Macdonald [21] (in our paper the letter $\Lambda$ is reserved for a different ring). Consider the Grothendieck ring $\mathop{\rm Grot}\nolimits(R)$ of the category of finite dimensional right $A(R)$-comodules. In the semisimple case one can use a quantum version of the Schur-Weyl duality to obtain a ring homomorphism $\mathop{\rm Sym}\nolimits\to\mathop{\rm Grot}\nolimits(R)$ under which each Schur function $s_{\lambda}$ is sent either to 0 or to the class of a simple comodule.

Since $\mathop{\rm Sym}\nolimits$ is a polynomial ring in a countable set of indeterminates, a homomorphism $\varphi:\mathop{\rm Sym}\nolimits\to\mathop{\rm Grot}\nolimits(R)$ can be easily constructed in the case of arbitrary $q$ by specifying its values on the generators. The main obstacle we encounter is to show that the element $\varphi(s_{\lambda})\in\mathop{\rm Grot}\nolimits(R)$ is positive in the sense that $\varphi(s_{\lambda})$ is the class of an actual comodule $V^{\lambda}$, in general not defined uniquely, of course. What is needed here can be reformulated in terms of the Grothendieck group of the category of finite dimensional $\mathop{\rm End}\nolimits_{{\cal H}_{n}}\mskip-5.0muX$-modules where $X=V^{\otimes n}$ with the ${\cal H}_{n}$-module structure arising from $R$. The necessary property is stated in Corollary 2.8, and section 2 is devoted to its proof. A key role is played by a version of the decomposition map which provides a bridge between the Grothendieck groups in the semisimple and nonsemisimple cases.

Total positivity of the sequence \bigl{(}\dim{{S}}_{n}(V,R)\bigr{)} is an immediate consequence of positivity of the images of the Schur functions under $\varphi$. Once it is known, we can invoke the analytic result of [1] and [12]. Actually it will be shown in section 3 that rationality of the Hilbert series can be explained by purely algebraic arguments, and the remainder of the proof is then much shorter than in general. In this way we present a selfcontained proof of Theorem 3.8.

Under the same assumption about $R$ it will be shown in section 4 that the class of $V^{\otimes n}$ in the Grothendieck group of the category of finite dimensional ${\cal H}_{n}$-modules is completely determined by the Hilbert series of ${{S}}(V,R)$. Moreover, we describe in Theorem 4.5 a certain element $\mathop{\rm ch}\nolimits(V^{\otimes n})\in\mathop{\rm Sym}\nolimits$ which contains full information about this class $[V^{\otimes n}]$. However, it is not clear whether $V^{\otimes n}$ can always be determined as an ${\cal H}_{n}$-module up to isomorphism.

If the algebra $\Lambda(V,R)$ is finite dimensional and $R$ satisfies the trivial source condition in the sense that for each $n>0$ the indecomposable ${\cal H}_{n}$-module direct summands of $V^{\otimes n}$ are induced from the trivial $1$-dimensional representations of parabolic subalgebras of ${\cal H}_{n}$, then our results are much more complete. Indeed, the ${\cal H}_{n}$-module $V^{\otimes n}$ is described in Theorem 6.1. As a consequence, in this case the algebra $A_{n}(R)^{*}$ dual to the subcoalgebra $A_{n}(R)\subset A(R)$ is Morita equivalent to the $q$-Schur algebra of Dipper and James $S_{q}(r,n)$ [8] where $r=r_{0}$ is the rank of $R$ (we denote by $A_{n}(R)$ the degree $n$ homogeneous component of $A(R)$). Therefore the category of $A_{n}(R)$-comodules is equivalent to the well studied category of $S_{q}(r,n)$-modules. In particular, this category depends only on $q$, $r$, and $n$, but not on $R$ itself. It is a highest weight category (see Donkin [9]).

It has been known for a long time that $A_{n}(R)^{*}\cong S_{q}(r,n)$ for many different quantizations of the semigroup of $r\times r$ matrices. This phenomenon was first observed by Du, Parshall and Wang [11] in the case of Takeuchi’s 2-parameter family of deformations. In an equivalent formulation, two bialgebras $A(R)$ and $A(R^{\prime})$ in this family are isomorphic as coalgebras whenever the corresponding parameters satisfy a certain relation, and then their corepresentation categories are obviously equivalent. As is seen from [11, (2.7)] it was not clear at that time whether these two categories are monoidally equivalent. On the level of Hopf envelopes a general result on braided monoidal equivalence was obtained later by Phùng Hô Hai in the semisimple case [18].

We will use Theorem 6.1 to strengthen two results from the previous paper [24]. Keeping the previous assumption about $R$, let $R^{\prime}$ be a second Hecke symmetry satisfying the same conditions. Theorem 6.3 states that there is a braided monoidal equivalence between the categories of $A(R)$-comodules and $A(R^{\prime})$-comodules provided that the two Hecke symmetries have the same parameter $q$ and the same rank $r$. This equivalence is obtained by cotensoring right $A(R)$-comodules with the bicomodule algebra $A(R^{\prime},R)$ (see [24, Th. 7.2]). By Theorem 6.4 the graded algebra $A(R^{\prime},R)$ is Gorenstein under similar assumptions, this time the equality of ranks is not required.

The trivial source indecomposable ${\cal H}_{n}$-modules are known as the Young modules [7]. As shown in [8], they are parametrized by partitions of $n$. Arbitrary indecomposable ${\cal H}_{n}$-modules with a 1-dimensional source may be called signed Young modules as in the case of representations of symmetric groups [10], [14]. However, an earlier text of Donkin [9] uses this term in a more restricted sense. There may be more such modules than partitions of $n$, and then the ${\cal H}_{n}$-modules are not distinguished by their images in $\mathop{\rm Sym}\nolimits$. Because of this we cannot generalize Theorem 6.1 to Hecke symmetries satisfying the 1-dimensional source condition.

In the semisimple case the algebras $A(R)$ and $A(R^{\prime},R)$ also have rational Hilbert series. Moreover, Phùng Hô Hai gives a formula for $H_{A(R^{\prime}\!,R)}$ in terms of $H_{{{S}}(V,R)}$ and $H_{{{S}}(V^{\prime}\!,R^{\prime})}$ [17, Th. 3.1]. In section 5 these results are extended to the case of arbitrary $q$ under the assumption that both $R$ and $R^{\prime}$ satisfy the $1$-dimensional source condition.

1. Preliminaries

We fix an arbitrary field $\mathchar 1404\relax$. Unless specified otherwise algebras and coalgebras will be considered over $\mathchar 1404\relax$. Let $V$ be a finite dimensional vector space over $\mathchar 1404\relax$. A Hecke symmetry on $V$ is a linear operator $R:V\otimes V\to V\otimes V$ satisfying the braid equation

[TABLE]

and the quadratic Hecke relation

[TABLE]

Denote by ${\cal H}_{n}(q)$ the Hecke algebra of type $A_{n-1}$ with the same parameter $q$ as in the quadratic relation imposed on $R$. Since $R$ and $q$ will generally be fixed, we do not indicate $q$ in the notation ${\cal H}_{n}$ when there is no danger of confusion. The algebra ${\cal H}_{n}$ is generated by $n-1$ elements $T_{1},\ldots,T_{n-1}$ subject to the defining relations

[TABLE]

Let ${{S}}_{n}$ be the symmetric group of permutations of the set $\{1,\ldots,n\}$. It is generated by basic transpositions $\tau_{i}=(i,i+1)$, $\,0<i<n$. Denote by $\ell(\sigma)$ the length of a permutation $\sigma\in{{S}}_{n}$ with respect to these generators. There is a standard basis $\{T_{\sigma}\mid\sigma\in{{S}}_{n}\}$ of ${\cal H}_{n}$ characterized by the properties that $T_{\tau_{i}}=T_{i}$ for each $i$ and $T_{\pi\sigma}=T_{\pi}T_{\sigma}$ for $\pi,\sigma\in{{S}}_{n}$ whenever $\ell(\pi\sigma)=\ell(\pi)+\ell(\sigma)$. We adopt the convention that ${\cal H}_{0}={\cal H}_{1}=\mathchar 1404\relax$.

The Hecke symmetry $R$ gives rise to a representation of ${\cal H}_{n}$ in the $n$th tensor power of $V$ such that $T_{i}$ acts on $V^{\otimes n}$ as the linear operator

[TABLE]

In this way $V^{\otimes n}$ becomes a left ${\cal H}_{n}$-module.

Denote by $A(R)$ the $R$-matrix bialgebra. It decomposes as a direct sum of subcoalgebras

[TABLE]

where $A_{n}(R)$ is the coalgebra dual to the finite dimensional algebra $\mathop{\rm End}\nolimits_{{\cal H}_{n}}\mskip-5.0muV^{\otimes n}$.

Let $\mathop{\rm Coend}\nolimits V$ be the coalgebra dual to $\mathop{\rm End}\nolimits\nolimits_{\mskip 2.0mu\mathchar 1404\relax}\mskip-2.0muV$. For each $n\geq 0$ we may identify $(\mathop{\rm Coend}\nolimits V)^{\otimes n}$ with the dual of the algebra $(\mathop{\rm End}\nolimits\nolimits_{\mskip 2.0mu\mathchar 1404\relax}\mskip-2.0muV)^{\otimes n}\cong\mathop{\rm End}\nolimits\nolimits_{\mskip 2.0mu\mathchar 1404\relax}\mskip-2.0muV^{\otimes n}$. Since $\mathop{\rm End}\nolimits_{{\cal H}_{n}}\mskip-5.0muV^{\otimes n}$ is a subalgebra of $\mathop{\rm End}\nolimits\nolimits_{\mskip 2.0mu\mathchar 1404\relax}\mskip-2.0muV^{\otimes n}$, we have

[TABLE]

where $I_{n}=(\mathop{\rm End}\nolimits_{{\cal H}_{n}}\mskip-5.0muV^{\otimes n})^{\perp}=\{f\in(\mathop{\rm Coend}\nolimits V)^{\otimes n}\mid\langle f,\,\mathop{\rm End}\nolimits_{{\cal H}_{n}}\mskip-5.0muV^{\otimes n}\rangle=0\}$ is a coideal of $(\mathop{\rm Coend}\nolimits V)^{\otimes n}$. Clearly $I_{n}=0$ for $n=0,\mskip 1.0mu1$. For $n>1$ there is an equality

[TABLE]

where $E_{i}^{(n)}$ stands for the centralizer of $R_{i}^{(n)}$ in $\mathop{\rm End}\nolimits\nolimits_{\mskip 2.0mu\mathchar 1404\relax}\mskip-2.0muV^{\otimes n}$. Since

[TABLE]

for each $i$, we get

[TABLE]

This shows that $I=\bigoplus_{n=0}^{\infty}I_{n}$ is an ideal of the tensor algebra

[TABLE]

generated by the homogeneous component $I_{2}$ of degree 2. By an earlier observation $I$ is also a coideal. Therefore $A(R)\cong{{T}}(\mathop{\rm Coend}\nolimits V)/I$ gets the structure of a factor bialgebra of ${{T}}(\mathop{\rm Coend}\nolimits V)$. This bialgebra coacts on $V$ universally with respect to the property that the induced coaction on $V^{\otimes 2}$ commutes with $R$. As observed in [20], this property characterizes the bialgebra arising from the FRT construction.

For an associative algebra ${{A}}$ over some field we denote by $\mathop{\rm Grot}\nolimits{{A}}$ the Grothendieck group of the category of finite dimensional left ${{A}}$-modules. To each finite dimensional left ${{A}}$-module $X$ there corresponds an element $[X]\in\mathop{\rm Grot}\nolimits{{A}}$, and to each short exact sequence $0\to X^{\prime}\to X\to X^{\prime\prime}\to 0$ of finite dimensional left ${{A}}$-modules there corresponds a relation $[X]=[X^{\prime}]+[X^{\prime\prime}]$ in this group. The elements corresponding to finite dimensional left ${{A}}$-modules form a subsemigroup of $\mathop{\rm Grot}\nolimits{{A}}$. Given $\xi\in\mathop{\rm Grot}\nolimits{{A}}$, we write $\xi\geq 0$ if $\xi$ lies in that subsemigroup, i.e., if $\xi=[X]$ for some finite dimensional left ${{A}}$-module $X$.

By the definition we have given above

[TABLE]

Therefore right $A_{n}(R)$-comodules may be identified with left modules for the algebra $\mathop{\rm End}\nolimits_{{\cal H}_{n}}\mskip-5.0muV^{\otimes n}$. The Grothendieck group $\mathop{\rm Grot}\nolimits_{n}(R)$ of the category of finite dimensional right $A_{n}(R)$-comodules is identified with the group $\,\mathop{\rm Grot}\nolimits(\mathop{\rm End}\nolimits_{{\cal H}_{n}}\mskip-5.0muV^{\otimes n})$. The Grothendieck group of the category of finite dimensional right $A(R)$-comodules is the direct sum

[TABLE]

Moreover, $\mathop{\rm Grot}\nolimits(R)$ is a graded ring with respect to the multiplication induced by tensor products of comodules.

The algebras ${{S}}(V,R)$ and $\Lambda(V,R)$ are defined as the factor algebras of the tensor algebra ${{T}}(V)=\bigoplus_{n=0}^{\infty}\,V^{\otimes n}$ by the ideals generated, respectively, by the subspaces

[TABLE]

These ideals are stable under the coaction of $A(R)$ on ${{T}}(V)$, and therefore ${{S}}(V,R)$ and $\Lambda(V,R)$ are right $A(R)$-comodule algebras in a natural way. The homogeneous components ${{S}}_{n}(V,R)$ and $\Lambda_{n}(V,R)$ of these algebras are right $A_{n}(R)$-comodules for each $n\geq 0$.

A composition of $n$ is any finite sequences of positive integers $\lambda=(\lambda_{1},\ldots,\lambda_{k})$ with $|\lambda|=n$ where the weight of $\lambda$ is defined as $|\lambda|=\sum\lambda_{i}$. The length of $\lambda$ is the number $\ell(\lambda)=k$ of its parts $\lambda_{i}$. As is done customarily, we extend $\lambda$ by putting $\lambda_{i}=0$ for $i>\ell(\lambda)$. If $\lambda_{i}\geq\lambda_{i+1}$ for all $i$, then $\lambda$ is called a partition of $n$. Denote by ${\cal P}(n)$ the set of all partitions of $n$ and by ${\cal P}$ the disjoint union of the sets ${\cal P}(0),\,{\cal P}(1),\,{\cal P}(2),\ldots$ where ${\cal P}(0)$ is regarded as a single element set consisting of the zero partition [math].

We denote by $\,\mathop{\rm Sym}\nolimits\,$ the ring of symmetric functions in a countable set of commuting indeterminates $x_{1},x_{2}\mskip 1.0mu,\ldots$ defined as in Macdonald [21]. Thus, if $\,\mathop{\rm Sym}\nolimits(r)\,$ stands for the subring of symmetric polynomials in the ring ${{Z}}[x_{1},\ldots,x_{r}]$, then $\mathop{\rm Sym}\nolimits$ is the limit of the inverse system

[TABLE]

in the category of graded rings. If $u\in\mathop{\rm Sym}\nolimits$ and $\alpha=(\alpha_{1},\ldots,\alpha_{r})\in K^{r}$ where $K$ is any commutative ring, then $u(\alpha)\in K$ is defined as the value at $\alpha$ of the polynomial obtained by projecting $u$ to $\mathop{\rm Sym}\nolimits(r)$.

We conform to standard notation in regard to several families of symmetric functions [21]. The elementary and complete symmetric functions will be $e_{n}$ and $h_{n}$ ($n=0,1,\ldots\,$) with $e_{0}=h_{0}=1$. For each partition $\lambda=(\lambda_{1},\ldots,\lambda_{k})$ one defines $e_{\lambda}=e_{\lambda_{1}}\!\cdots e_{\lambda_{k}}$ and $h_{\lambda}=h_{\lambda_{1}}\!\cdots h_{\lambda_{k}}$. The monomial functions $m_{\lambda}$ and the Schur functions $s_{\lambda}$ labelled by partitions $\lambda$ form two other well known ${{Z}}$-bases of $\mathop{\rm Sym}\nolimits$. On each homogeneous component $\mathop{\rm Sym}\nolimits_{n}$ of the graded ring $\mathop{\rm Sym}\nolimits$ there is a standard scalar product $\langle\cdot\,,\cdot\rangle$ with respect to which $\{s_{\lambda}\mid\lambda\in{\cal P}(n)\}$ is an orthonormal basis.

To each composition $\lambda$ of $n$ there corresponds a parabolic subalgebra ${\cal H}_{\lambda}={\cal H}_{\lambda}(q)$ of the Hecke algebra ${\cal H}_{n}={\cal H}_{n}(q)$. It is generated by the set $\{T_{i}\mid i\in{\cal I}_{\lambda}\}$ where

[TABLE]

and has a basis $\{T_{\sigma}\mid\sigma\in{{S}}_{\lambda}\}$ where ${{S}}_{\lambda}$ is the Young subgroup of ${{S}}_{n}$ generated by $\{\tau_{i}\mid i\in{\cal I}_{\lambda}\}$.

Each homomorphism $\chi:{\cal H}_{\lambda}\to\mathchar 1404\relax$ is completely determined by its values on the generators. It follows from the Hecke relations that $\chi(T_{i})\in\{-1,q\}$ for each $i\in{\cal I}_{\lambda}$. If both $i$ and $i+1$ lie in ${\cal I}_{\lambda}$ then $\chi(T_{i})=\chi(T_{i+1})$ by the braid relations. In other words, $\chi$ is constant on each of the $\ell(\lambda)$ contiguous (possibly empty) segments of lengths $\lambda_{i}-1$, $\,i=1,\ldots,\ell(\lambda)$, which comprise the set ${\cal I}_{\lambda}$.

If $\nu$ is a composition of $n$ obtained from $\lambda$ by permuting its components in an arbitrary order, then the two subalgebras ${\cal H}_{\lambda}$ and ${\cal H}_{\nu}$ are conjugate by an inner automorphism of ${\cal H}_{n}$. In this case the induction functors from ${\cal H}_{\lambda}$ and from ${\cal H}_{\nu}$ produce isomorphic ${\cal H}_{n}$-modules. Given a homomorphism $\chi:{\cal H}_{\lambda}\to\mathchar 1404\relax$, it is possible to pass to an equivalent homomorphism $\chi^{\prime}:{\cal H}_{\nu}\to\mathchar 1404\relax$ such that all segments on which $\chi^{\prime}$ takes value $q$ precede all segments on which $\chi^{\prime}$ takes value $-1$, and any pair of segments on which $\chi^{\prime}$ takes the same value follow in nonincreasing order of their lengths.

When forming the ${\cal H}_{n}$-modules induced from 1-dimensional representations of parabolic subalgebras, it suffices to consider only the pairs $({\cal H}_{\nu},\chi^{\prime})$ satisfying the previous conditions. This leads to a parametrization of such modules by the set

[TABLE]

Each pair $(\lambda,\mu)\in{\cal P}^{2}(n)$ determines a composition

[TABLE]

We will denote by ${\cal H}_{\lambda,\mskip 1.0mu\mu}$ the corresponding parabolic subalgebra of ${\cal H}_{n}$, by ${{S}}_{\lambda,\mskip 1.0mu\mu}$ the corresponding Young subgroup of ${{S}}_{n}$, and by ${\cal I}_{\lambda,\mskip 1.0mu\mu}$ the index set for the generators $T_{i}\in{\cal H}_{\lambda,\mskip 1.0mu\mu}$ and $\tau_{i}\in{{S}}_{\lambda,\mskip 1.0mu\mu}$. Then

[TABLE]

Note that ${\cal H}_{\lambda,\mskip 1.0mu\mu}\cong{\cal H}_{\lambda}\otimes{\cal H}_{\mu}$. Define a homomorphism $\chi_{\lambda,\mskip 1.0mu\mu}:{\cal H}_{\lambda,\mskip 1.0mu\mu}\to\mathchar 1404\relax$ by the rule

[TABLE]

and denote by $\mathchar 1404\relax_{\lambda,\mskip 1.0mu\mu}$ the corresponding 1-dimensional ${\cal H}_{\lambda,\mskip 1.0mu\mu}$-module.

If $\mu=0$, then ${\cal H}_{\lambda,\mskip 1.0mu\mu}={\cal H}_{\lambda}$ and $\mathchar 1404\relax_{\lambda,\mskip 1.0mu\mu}$ is the trivial ${\cal H}_{\lambda}$-module $\mathchar 1404\relax_{\rm triv}$ on which each generator $T_{i}\in{\cal H}_{\lambda}$ operates as multiplication by $q$. If $\lambda=0$, then ${\cal H}_{\lambda,\mskip 1.0mu\mu}={\cal H}_{\mu}$ and $\mathchar 1404\relax_{\lambda,\mskip 1.0mu\mu}$ is the alternating ${\cal H}_{\mu}$-module $\mathchar 1404\relax_{\rm alt}$ on which each $T_{i}\in{\cal H}_{\mu}$ operates as multiplication by $-1$.

From the preceding discussion it follows that an indecomposable left ${\cal H}_{n}$-module has a $1$-dimensional (respectively, trivial) source if and only if it is isomorphic to a direct summand of the induced module ${\cal H}_{n}\otimes_{{\cal H}_{\lambda,\mu}}\mathchar 1404\relax_{\lambda,\mskip 1.0mu\mu}$ (respectively, ${\cal H}_{n}\otimes_{{\cal H}_{\lambda}}\mathchar 1404\relax_{\rm triv}$) for some $(\lambda,\mu)\in{\cal P}^{2}(n)$ (respectively, $\lambda\in{\cal P}(n)$).

The Specht ${\cal H}_{n}$-modules $S^{\lambda}$ labelled by partitions $\lambda\in{\cal P}(n)$ were constructed by Dipper and James [6]. We will use this notation for left modules and sometimes also for right modules as in [6]. The dimension of $S^{\lambda}$ depends neither on the field $\mathchar 1404\relax$ nor on the parameter $q$. In particular, it is the same as the dimension of the respective Specht module for the symmetric group ${{S}}_{n}$.

The Hecke algebras ${\cal H}_{0},{\cal H}_{1},{\cal H}_{2},\ldots$ are all semisimple if and only if $[n]_{q}\neq 0$ for all $n>0$ where

[TABLE]

We now recall briefly what happens in the semisimple case. The simple ${\cal H}_{n}$-modules are precisely the Specht modules. Moreover, each $S^{\lambda}$ is absolutely irreducible, so that $\mathop{\rm End}\nolimits_{{\cal H}_{n}}\!S^{\lambda}\cong\mathchar 1404\relax$. By semisimplicity of ${\cal H}_{n}$ an arbitrary ${\cal H}_{n}$-module $N$ is a direct sum of its isotypic components, and the isotypic component of type $S^{\lambda}$ can be expressed as $\mathop{\rm Hom}\nolimits_{{\cal H}_{n}}(S^{\lambda}\!,N)\otimes S^{\lambda}$. Taking $N=V^{\otimes n}\!$, we get

[TABLE]

It follows from this decomposition that

[TABLE]

Thus the algebra $\mathop{\rm End}\nolimits_{{\cal H}_{n}}\mskip-5.0muV^{\otimes n}$ is semisimple with $\{V_{R}^{\lambda}\mid\lambda\in{\cal P}(n),\ V_{R}^{\lambda}\neq 0\}$ being a full set of pairwise nonisomorphic simple left modules. Note that $V_{R}^{\lambda}\neq 0$ if and only if $S^{\lambda}$ embeds in $V^{\otimes n}$ as an ${\cal H}_{n}$-submodule. We see that all right $A(R)$-comodules are semisimple, which means that $A(R)$ is a cosemisimple bialgebra. Moreover,

[TABLE]

is a full set of pairwise nonisomorphic simple right $A(R)$-comodules. Their isomorphism classes $[V^{\lambda}]$ form a ${{Z}}$-basis of the Grothendieck group $\mathop{\rm Grot}\nolimits(R)$. Note also that $\dim S^{\lambda}$ is the multiplicity of $V_{R}^{\lambda}$ as an $A(R)$-comodule summand of $V^{\otimes n}$, while $\dim V_{R}^{\lambda}$ is the multiplicity of $S^{\lambda}$ as an ${\cal H}_{n}$-module summand of $V^{\otimes n}$.

Let ${\cal H}_{m,n}$ be the subalgebra of ${\cal H}_{m+n}$ generated by $\{T_{i}\mid 0<i<m+n,\ i\neq\nobreak m\}$. In other words, ${\cal H}_{m,n}={\cal H}_{(m),(n)}$ for partitions $(m),(n)$ of length 1. Denote by $\mathop{\rm ind}\nolimits_{m,n}^{m+n}$ the induction functor from ${\cal H}_{m,n}$ to ${\cal H}_{m+n}$. There is a graded ring structure on the direct sum of Grothendieck groups $\bigoplus_{k=0}^{\infty}\mathop{\rm Grot}\nolimits{\cal H}_{k}$ defined by the rule

[TABLE]

whenever $X$ is an ${\cal H}_{m}$-module and $Y$ an ${\cal H}_{n}$-module, both of finite dimension. Here $X\otimes Y$ is viewed as an ${\cal H}_{m,n}$-module by means of the canonical isomorphism ${\cal H}_{m,n}\cong{\cal H}_{m}\otimes{\cal H}_{n}$. In this ring

[TABLE]

with the Littlewood-Richardson coefficients $c_{\mu\nu}^{\lambda}$ which also occur as structure constants for the multiplication in the ring of symmetric functions:

[TABLE]

This means that there is an isomorphism of graded rings

[TABLE]

under which $s_{\lambda}\in\mathop{\rm Sym}\nolimits$ corresponds to $[S^{\lambda}]\in\mathop{\rm Grot}\nolimits{\cal H}_{|\lambda|}$. Conceptual explanation of this isomorphism is provided by Zelevinsky’s approach [26]. With some additional structure the direct sum of the groups $\mathop{\rm Grot}\nolimits{\cal H}_{k}$ satisfies the axioms of a connected positive selfadjoint Hopf algebra over ${{Z}}$, and it contains only one irreducible primitive element. It was proved in [26] that such a Hopf algebra is unique up to isomorphism and is isomorphic to the ring of symmetric functions.

Lemma 1.1. For each $(\lambda,\mu)\in{\cal P}^{2}(n)$ the symmetric function $h_{\lambda}e_{\mu}$ maps to the class of the induced module ${\cal H}_{n}\otimes_{{\cal H}_{\lambda,\mu}}\mathchar 1404\relax_{\lambda,\mskip 1.0mu\mu}$ in the Grothendieck group $\mathop{\rm Grot}\nolimits{\cal H}_{n}$. As a consequence,* the following relations hold in this group*:**

[TABLE]

where $\mathchar 1404\relax_{i,\mskip 1.0mun-i}$ is the $1$-dimensional ${\cal H}_{i,\mskip 1.0mun-i}$-module associated with the pair of partitions \bigl{(}(i),(n-i)\bigr{)}\in{\cal P}^{2}(n) and $K_{\lambda\mu}$ are the Kostka numbers.

Proof. Since for each $p>0$ the symmetric functions $h_{p}=s_{(p)}$ and $e_{p}=s_{(1^{p})}$ map to the classes of the Specht modules $S^{(p)}=\mathchar 1404\relax_{\rm triv}$ and $S^{(1^{p})}=\mathchar 1404\relax_{\rm alt}$ in the group $\mathop{\rm Grot}\nolimits{\cal H}_{p}$, it follows from the definition of the multiplication in the direct sum of the groups $\mathop{\rm Grot}\nolimits{\cal H}_{k}$ that $h_{\lambda}$ and $e_{\lambda}$, for any $\lambda\in{\cal P}$, map to the classes of induced modules ${\cal H}_{|\lambda|}\otimes_{{\cal H}_{\lambda}}\mathchar 1404\relax_{\rm triv}$ and ${\cal H}_{|\lambda|}\otimes_{{\cal H}_{\lambda}}\mathchar 1404\relax_{\rm alt}$ in the group $\mathop{\rm Grot}\nolimits{\cal H}_{|\lambda|}$. Hence

[TABLE]

for $(\lambda,\mu)\in{\cal P}^{2}(n)$. The required equalities in the group $\mathop{\rm Grot}\nolimits{\cal H}_{n}$ are now immediate consequences of the well known equalities

[TABLE]

in the group $\mathop{\rm Sym}\nolimits_{n}$ (see [21, Ch. I]).   $\mathchar 9219\relax$

In the semisimple case tensor products of simple $A(R)$-comodules are computed easily. If $\mu\in{\cal P}(m)$ and $\nu\in{\cal P}(n)$, then

[TABLE]

It follows that for each $\lambda\in{\cal P}(m+n)$ the multiplicity of $V_{R}^{\lambda}$ in $V_{R}^{\mu}\otimes V_{R}^{\nu}$ equals the multiplicity of the simple ${\cal H}_{m+n}$-module $S^{\lambda}$ in $\,\mathop{\rm ind}\nolimits_{m,n}^{m+n}S^{\mu}\otimes S^{\nu}$. Thus

[TABLE]

and so there is a surjective ring homomorphism $\varphi:\mathop{\rm Sym}\nolimits\to\mathop{\rm Grot}\nolimits(R)$ given by the assignments $s_{\lambda}\mapsto[V_{R}^{\lambda}]$, $\,\lambda\in{\cal P}$. Let

[TABLE]

be the set of $(r_{0},r_{1})$-hook partitions. It was proved by Phùng Hô Hai [16] that $V_{R}^{\lambda}\neq 0$ if and only if $\lambda\in\Gamma(r_{0},r_{1})$ where $(r_{0},r_{1})$ is the birank of $R$. Hence the classes $[V_{R}^{\lambda}]$ with $\lambda\in\Gamma(r_{0},r_{1})$ form a ${{Z}}$-basis of $\mathop{\rm Grot}\nolimits(R)$ and $\mathop{\rm Ker}\varphi$ coincides with the ${{Z}}$-linear span of $\{s_{\lambda}\mid\lambda\notin\Gamma(r_{0},r_{1})\}$.

In the present paper it will be shown that some features of this situation extend to the nonsemisimple case provided that $R$ satisfies the 1-dimensional source condition.

2. The decomposition map

The decomposition map is a standard tool in the modular representation theory of finite groups. More generally, such a map can be defined in the following situation (see, e.g., [13, 7.4.3]). Suppose that $O$ is a discrete valuation ring with residue field $\mathchar 1404\relax$ and the field of fractions $Q$. Let $A$ be an associative unital algebra over $O$ whose underlying $O$-module is free of finite rank. Then $A_{\mskip 0.5mu\mathchar 1404\relax}=A\otimes_{O}\mathchar 1404\relax$ and $A_{Q}=A\otimes_{O}Q$ are finite dimensional algebras over the respective fields. By an $A$-lattice we mean any finitely generated $O$-free $A$-module (a finitely generated $O$-module is free if and only if it is torsionfree). The decomposition map

[TABLE]

is a homomorphism of groups characterized by the property that $d([L_{Q}])=[L_{\mskip 1.0mu\mathchar 1404\relax}\mskip 0.5mu]$ for each left $A$-lattice $L$ where $L_{Q}=L\otimes_{O}Q$ and $L_{\mskip 1.0mu\mathchar 1404\relax}=L\otimes_{O}\mathchar 1404\relax$. This map is well-defined since each $A_{Q}$-module of finite dimension over $Q$ is isomorphic to $L_{Q}$ for some $A$-lattice $L$, and the image of $L_{\mskip 1.0mu\mathchar 1404\relax}$ in $\mathop{\rm Grot}\nolimits A_{\mskip 0.5mu\mathchar 1404\relax}$ does not depend on the choice of $L$.

Let $z$ be an invertible element of $O$ and ${\cal H}_{n}(z)$ the Hecke algebra of type $A_{n-1}$ with parameter $z$ over the ring $O$. If $q$ is the image of $z$ in $\mathchar 1404\relax$, then ${\cal H}_{n}(z)_{\mathchar 1404\relax}\cong{\cal H}_{n}(q)$. We will assume that

[TABLE]

This can be achieved, e.g., by taking $z$ to be an indeterminate and $O$ the localization of the polynomial ring $\mathchar 1404\relax[z]$ at its maximal ideal generated by $z-q$. Then ${\cal H}_{n}(z)_{Q}$ is a semisimple Hecke algebra of type $A_{n-1}$ over the field $Q$. By completing $O$ we may also assume that $O$ is a complete discrete valuation ring.

We will denote by $T_{1},\ldots,T_{n-1}$ the canonical generators of ${\cal H}_{n}(z)$ and also their canonical images in either ${\cal H}_{n}(q)$ or ${\cal H}_{n}(z)_{Q}$. For $(\lambda,\mu)\in{\cal P}^{2}(n)$ we have a parabolic subalgebra ${\cal H}_{\lambda,\mskip 1.0mu\mu}(z)$ generated by $\{T_{i}\mid i\in{\cal I}_{\lambda,\mskip 1.0mu\mu}\}$ (see section 1). Define a homomorphism of $O$-algebras $\chi_{\lambda,\mskip 1.0mu\mu}:{\cal H}_{\lambda,\mskip 1.0mu\mu}(z)\to O$ by the rule

[TABLE]

and let $O_{\lambda,\mskip 1.0mu\mu}$ be $O$ with the ${\cal H}_{\lambda,\mskip 1.0mu\mu}(z)$-module structure arising from $\chi_{\lambda,\mskip 1.0mu\mu}$. Let $Q_{\lambda,\mskip 1.0mu\mu}$ be the 1-dimensional ${\cal H}_{\lambda,\mskip 1.0mu\mu}(z)_{Q}$-module defined similarly. We will write ${\cal H}_{\lambda}(z)$, $O_{\rm triv}$, and $Q_{\rm triv}$ instead of ${\cal H}_{\lambda,\mskip 1.0mu\mu}(z)$, $O_{\lambda,\mskip 1.0mu\mu}$, and $Q_{\lambda,\mskip 1.0mu\mu}$ when $\lambda\in{\cal P}(n)$ and $\mu=0$.

Notation. Denote by ${Rep}^{1}$ (respectively, by ${Triv}$) the class of all ${\cal H}_{n}(z)$-lattices isomorphic to direct summands of finite direct sums of the induced modules

[TABLE]

for various $(\lambda,\mu)\in{\cal P}^{2}(n)$ (respectively, $\lambda\in{\cal P}(n)$).

Note that $M^{\lambda}=M^{\lambda,\mskip 1.0mu0}$. Since ${\cal H}_{n}(z)$ is a free ${\cal H}_{\lambda,\mskip 1.0mu\mu}(z)$-module with respect to the action by right multiplications, $M^{\lambda,\mskip 1.0mu\mu}$ is an ${\cal H}_{n}(z)$-lattice. The functor $?\otimes_{O}\mathchar 1404\relax$ takes $M^{\lambda,\mskip 1.0mu\mu}$ to

[TABLE]

It follows that all indecomposable direct summands of the ${\cal H}_{n}(q)$-module $M_{\mskip 1.0mu\mathchar 1404\relax}$ have a 1-dimensional (respectively, trivial) source whenever $M\in{Rep}^{1}$ (respectively, $M\in{Triv}$).

Lemma 2.1. Let $M,N\in{Rep}^{1}$. If $q=-1$ assume that $M,N\in{Triv}$. Then

[TABLE]

Hence $\dim\nolimits_{\mskip 2.0mu\mathchar 1404\relax}\mathop{\rm Hom}\nolimits_{{\cal H}_{n}(q)}(N_{\mskip 1.0mu\mathchar 1404\relax},M_{\mskip 1.0mu\mathchar 1404\relax})=\dim_{Q}\mathop{\rm Hom}\nolimits_{{\cal H}_{n}(z)_{Q}}(N_{Q},M_{Q})$.

Proof. The canonical maps $k_{XY}:\mathop{\rm Hom}\nolimits_{{\cal H}_{n}(z)}(X,Y)\otimes_{O}\mathchar 1404\relax\to\mathop{\rm Hom}\nolimits_{{\cal H}_{n}(q)}(X_{\mathchar 1404\relax},Y_{\mathchar 1404\relax})$ defined for arbitrary ${\cal H}_{n}(z)$-modules $X$ and $Y$ give a natural transformation of two functors additive in each argument. If $X\cong X^{\prime}\oplus X^{\prime\prime}$ (respectively, $Y\cong Y^{\prime}\oplus Y^{\prime\prime}$), then bijectivity of $k_{XY}$ is equivalent to bijectivity of $k_{X^{\prime}Y}$ and $k_{X^{\prime\prime}Y}$ (respectively, $k_{XY^{\prime}}$ and $k_{XY^{\prime\prime}}$). By the conditions on $M$ and $N$ in the statement of Lemma 2.1 it suffices therefore to prove that $k_{NM}$ is bijective when $q\neq-1$ and $N=M^{\lambda,\mskip 1.0mu\mu}$, $M=M^{\rho,\mskip 1.0mu\theta}$ for some pairs $(\lambda,\mu),\,(\rho,\theta)\in{\cal P}^{2}(n)$ or when $q=-1$ and $N=M^{\lambda}$, $M=M^{\rho}$ with $\lambda,\rho\in{\cal P}(n)$.

Consider the case $q\neq-1$. Denote by ${\cal D}$ the set of distinguished representatives of the ${{S}}_{\lambda,\mskip 1.0mu\mu}\,$-$\,{{S}}_{\rho,\mskip 1.0mu\theta}$ double cosets in ${{S}}_{n}$. Then $M=\bigoplus_{\pi\in{\cal D}}\,M(\pi)$ where $M(\pi)$ is the ${\cal H}_{\lambda,\mskip 1.0mu\mu}(z)$-submodule of $M$ generated by the element $T_{\pi}\otimes 1\in M$. This is the Mackey decomposition of the induced module $M$ with respect to the parabolic subalgebra ${\cal H}_{\lambda,\mskip 1.0mu\mu}(z)$ (see [6, 2.7] and [13, 9.1.8]). For each $\pi\in{\cal D}$ let $\nu(\pi)$ be the composition of $n$ such that

[TABLE]

If $i\in{\cal I}_{\nu(\pi)}$, then $\ell(\tau_{i}\pi)>\ell(\pi)$ since $\pi$ is the shortest element in the coset ${{S}}_{\lambda,\mskip 1.0mu\mu}\,\pi$, and the equality $\tau_{i}=\pi\mskip 1.0mu\tau_{j}\pi^{-1}$ means that $i=\pi(j)$ and $i+1=\pi(j+1)$. Therefore $T_{i}T_{\pi}=T_{\tau_{i}\pi}=T_{\pi}T_{j}$ with $j=\pi^{-1}(i)\in{\cal I}_{\rho,\mskip 1.0mu\theta}$, i.e., $T_{\pi}^{-1}T_{i}T_{\pi}=T_{\pi^{-1}(i)}\in{\cal H}_{\rho,\mskip 1.0mu\theta}(z)$. By the Mackey formula

[TABLE]

where $O(\chi^{\prime}_{\pi})$ is $O$ regarded as an ${\cal H}_{\nu(\pi)}(z)$-module by means of the ring homomorphism $\chi^{\prime}_{\pi}:{\cal H}_{\nu(\pi)}(z)\to O$ such that

[TABLE]

Note that

[TABLE]

By the Frobenius reciprocity (see [6, 2.5, 2.6] and [13, 9.1.7])

[TABLE]

The $O$-module \mathop{\rm Hom}\nolimits_{{\cal H}_{\nu(\pi)}(z)}\bigl{(}O_{\lambda,\mskip 1.0mu\mu},O(\chi^{\prime}_{\pi})\bigr{)} is nonzero precisely when $\chi_{\lambda,\mskip 1.0mu\mu}$ agrees with $\chi^{\prime}_{\pi}$ on ${\cal H}_{\nu(\pi)}(z)$, in which case this module is isomorphic to $O$. It follows that $\mathop{\rm Hom}\nolimits_{{\cal H}_{n}(z)}(M,N)$ is a free $O$-module with a basis indexed by the set of those $\pi\in{\cal D}$ for which $\,{\cal I}_{\nu(\pi)}\cap{\cal I}_{\lambda,\mskip 1.0mu\mu}^{0}={\cal I}_{\nu(\pi)}\cap\pi({\cal I}_{\rho,\mskip 1.0mu\theta}^{0})$.

The respective homomorphisms $N\to M$ can be described explicitly as in [6, 3.4]. The vector space $\mathop{\rm Hom}\nolimits_{{\cal H}_{n}(q)}(N_{\mskip 1.0mu\mathchar 1404\relax},M_{\mskip 1.0mu\mathchar 1404\relax})$ has a similar description, and we see that the functor $?\otimes_{O}\mathchar 1404\relax$ takes the basic homomorphisms $N\to M$ to the basic homomorphisms $N_{\mskip 1.0mu\mathchar 1404\relax}\to M_{\mskip 1.0mu\mathchar 1404\relax}$. If $\mu=0$ and $\theta=0$, then the basic homomorphisms are parametrized by the whole set ${\cal D}$, and the previous arguments go through for $q=-1$ as well. This proves bijectivity of $k_{NM}$.

The second isomorphism in the statement of Lemma 2.1 can be explained in exactly the same way, but in fact it is almost obvious and holds more generally for arbitrary ${\cal H}_{n}(z)$-modules $M$ and $N$ with the only restriction that $N$ should be finitely generated. In the last assertion of Lemma 2.1 both dimensions are equal to the rank of the free $O$-module $\mathop{\rm Hom}\nolimits_{{\cal H}_{n}(z)}(N,M)$.   $\mathchar 9219\relax$

For the rest of this section with the exception of Lemma 2.7 we fix $M\in{Rep}^{1}$, and moreover we will assume that $M\in{Triv}$ when $q=-1$. The ring $A=\mathop{\rm End}\nolimits_{{\cal H}_{n}(z)}\mskip-2.0muM$ is an algebra over $O$ whose underlying $O$-module is free of finite rank. By Lemma 2.1 $A_{\mskip 0.5mu\mathchar 1404\relax}\cong\mathop{\rm End}\nolimits_{{\cal H}_{n}(q)}\mskip-2.0muM_{\mathchar 1404\relax}$ and $A_{Q}\cong\mathop{\rm End}\nolimits_{{\cal H}_{n}(z)_{Q}}\mskip-2.0muM_{Q}$. Thus we have the decomposition map

[TABLE]

Lemma 2.2. If $q\neq-1$ and $(\lambda,\mu)\in{\cal P}^{2}(n)$ then

[TABLE]

In particular,* $\,d(\,[\,Q_{\rm triv}\otimes_{{\cal H}_{\lambda}(z)_{Q}}\!M_{Q}\,]\,)=[\,\mathchar 1404\relax_{\rm triv}\otimes_{{\cal H}_{\lambda}(q)}M_{\mskip 1.0mu\mathchar 1404\relax}\,]\,$ for $\lambda\in{\cal P}(n),$ and this equality holds even when $q=-1$.*

Proof. Put $L=O_{\lambda,\mskip 1.0mu\mu}\otimes_{{\cal H}_{\lambda,\mu}(z)}M$. This is an $\mathop{\rm End}\nolimits_{{\cal H}_{n}(z)}\mskip-2.0muM$-module such that

[TABLE]

We will check that $L$ is $O$-free of finite rank, i.e. $L$ is a lattice. It will follow then that $d([L_{Q}])=[L_{\mskip 1.0mu\mathchar 1404\relax}]$ by the definition of $d$, and we will get the required equality.

Since the verifications can be done on direct summands, it suffices to consider the case when $M=M^{\rho,\mskip 1.0mu\theta}$ for some $(\rho,\theta)\in{\cal P}^{2}(n)$. Using the Mackey decomposition $M=\bigoplus_{\pi\in{\cal D}}M(\pi)$ with respect to ${\cal H}_{\lambda,\mskip 1.0mu\mu}(z)$ as in the proof of Lemma 2.1, we get $L=\bigoplus_{\pi\in{\cal D}}L(\pi)$ with

[TABLE]

where $I_{\pi}$ is the ideal of $O$ generated by $\{\chi_{\lambda,\mskip 1.0mu\mu}(h)-\chi^{\prime}_{\pi}(h)\mid h\in{\cal H}_{\nu(\pi)}(z)\}$. Since the $O$-algebra ${\cal H}_{\nu(\pi)}(z)$ is generated by $\{T_{i}\mid i\in{\cal I}_{\nu(\pi)}\}$, the ideal $I_{\pi}$ is generated by the elements

[TABLE]

If $\chi_{\lambda,\mskip 1.0mu\mu}$ agrees with $\chi^{\prime}_{\pi}$ on ${\cal H}_{\nu(\pi)}(z)$, then $I_{\pi}=0$. Otherwise $\chi_{\lambda,\mskip 1.0mu\mu}(T_{i})\neq\chi^{\prime}_{\pi}(T_{i})$ for at least one $i\in{\cal I}_{\nu(\pi)}$. Since $-1$ and $z$ are the only two possible values of the homomorphisms $\chi_{\lambda,\mskip 1.0mu\mu}$ and $\chi^{\prime}_{\pi}$ on the generators $T_{i}$, we get $I_{\pi}=(z+1)O$ in the latter case. The image of $z+1$ in the residue field $\mathchar 1404\relax$ of the local ring $O$ equals $q+1$. If $q\neq-1$, then $z+1$ is invertible in $O$, and so $I_{\pi}=O$. We see that $O/I_{\pi}$ equals either $O$ or [math] for each $\pi\in{\cal D}$ when $q\neq-1$, i.e. each $O$-module in the direct sum decomposition of $L$ is free of rank 1 or 0.

Suppose now that $M\in{Triv}$. In this case we may assume that $M=M^{\rho}$ for some $\rho\in{\cal P}(n)$, i.e. we can take $\theta=0$. Then $\chi^{\prime}_{\pi}$ is the trivial representation. If $\mu=0$, then $\chi_{\lambda,\mskip 1.0mu\mu}$ is the trivial representation as well, whence $I_{\pi}=0$ and $O/I_{\pi}\cong O$ for all $\pi$, even when $q=-1$.   $\mathchar 9219\relax$

Notation. For any ${\cal H}_{n}(q)$-module $X$ and $(\lambda,\mu)\in{\cal P}^{2}(n)$ put

[TABLE]

This is an $\mathop{\rm End}\nolimits_{{\cal H}_{n}(q)}\mskip-2.0muX$-submodule of $X$. In particular, we write

[TABLE]

when $\lambda=(i)$, $\mu=(n-i)$ for some $i=0,\ldots,n$, and we write

[TABLE]

when $\lambda\in{\cal P}(n)$, $\mu=0$.

The case $q=-1$ incurs technical complications, and we have to look deeper into the structure of induced modules. The conclusion of Lemma 2.2 is reformulated below in a form suitable for any $q$:

Lemma 2.3. We have $d(\,[\,Q_{\lambda,\mskip 1.0mu\mu}\otimes_{{\cal H}_{\lambda,\mu}(z)_{Q}}\!M_{Q}\,]\,)=[\,M_{\mskip 1.0mu\mathchar 1404\relax}/\Sigma_{\lambda,\mskip 1.0mu\mu}(M_{\mskip 1.0mu\mathchar 1404\relax})\,]$.

Proof. Put $L=O_{\lambda,\mskip 1.0mu\mu}\otimes_{{\cal H}_{\lambda,\mu}(z)}M$ and $A=\mathop{\rm End}\nolimits_{{\cal H}_{n}(z)}\mskip-2.0muM$. The assignment $m\mapsto 1\otimes m$ defines an epimorphism of $A$-modules $\psi:M\to L$. The induced map $\psi_{\mskip 1.0mu\mathchar 1404\relax}=\psi\otimes_{O}\mathchar 1404\relax$ is an epimorphism of $A_{\mskip 0.5mu\mathchar 1404\relax}$-modules $\,M_{\mskip 1.0mu\mathchar 1404\relax}\to L_{\mskip 1.0mu\mathchar 1404\relax}\cong\mathchar 1404\relax_{\lambda,\mskip 1.0mu\mu}\otimes_{{\cal H}_{\lambda,\mu}(q)}M_{\mskip 1.0mu\mathchar 1404\relax}\,$ such that

[TABLE]

Suppose that $q\neq-1$. It follows then from the relation $(T_{i}-q)(T_{i}+1)=0$ that the equality $T_{i}m=qm$ holds for an element $m\in M_{\mskip 1.0mu\mathchar 1404\relax}$ if and only if $m\in(T_{i}+1)M_{\mskip 1.0mu\mathchar 1404\relax}$. Hence $\,\mathop{\rm Ker}\psi_{\mskip 1.0mu\mathchar 1404\relax}=\Sigma_{\lambda,\mskip 1.0mu\mu}(M_{\mskip 1.0mu\mathchar 1404\relax})$, and

[TABLE]

The equality $d([L_{Q}])=[L_{\mskip 1.0mu\mathchar 1404\relax}]$ of Lemma 2.2 can be rewritten as in the statement of Lemma 2.3.

Suppose further that $q=-1$. In this case $L$ may fail to be $O$-free. The set $L^{0}$ of all elements of $L$ which have nonzero annihilators in $O$ is an $A$-submodule of $L$. The factor module ${\overline{L}}=L/L^{0}$ is $O$-torsionfree and ${\overline{L}}_{Q}\cong L_{Q}$. Hence ${\overline{L}}$ is a lattice, which yields $\,d([L_{Q}])=[\mkern 1.0mu{\overline{L}}_{\mathchar 1404\relax}\mkern 1.0mu]$.

We have ${\overline{L}}\cong M/M^{0}$ where $M^{0}=\psi^{-1}(L^{0})\subset M$. Consider $M_{\mskip 1.0mu\mathchar 1404\relax}^{0}=M^{0}\otimes_{O}\mathchar 1404\relax$. The canonical map $M_{\mskip 1.0mu\mathchar 1404\relax}^{0}\to M_{\mskip 1.0mu\mathchar 1404\relax}$ is injective since ${\overline{L}}$ is $O$-free. Thus $M_{\mskip 1.0mu\mathchar 1404\relax}^{0}$ is identified with an $A_{\mskip 0.5mu\mathchar 1404\relax}$-submodule of $M_{\mskip 1.0mu\mathchar 1404\relax}$. By the right exactness of the functor $?\otimes_{O}\mathchar 1404\relax$ we obtain ${\overline{L}}_{\mathchar 1404\relax}\cong M_{\mskip 1.0mu\mathchar 1404\relax}/M_{\mskip 1.0mu\mathchar 1404\relax}^{0}$, and it remains to show that $M_{\mskip 1.0mu\mathchar 1404\relax}^{0}=\Sigma_{\lambda,\mskip 1.0mu\mu}(M_{\mskip 1.0mu\mathchar 1404\relax})$.

Note that $M_{\mskip 1.0mu\mathchar 1404\relax}^{0}$ and $\Sigma_{\lambda,\mskip 1.0mu\mu}(M_{\mskip 1.0mu\mathchar 1404\relax})$ are evaluations at $M$ of two additive functors defined on the category of ${\cal H}_{n}(z)$-modules. Verification of the equality $M_{\mskip 1.0mu\mathchar 1404\relax}^{0}=\Sigma_{\lambda,\mskip 1.0mu\mu}(M_{\mskip 1.0mu\mathchar 1404\relax})$ can be done therefore on direct summands of $M$. Since $M\in{Triv}$, it suffices to consider the case when $M=M^{\rho}$ for some $\rho\in{\cal P}(n)$. Let us assume this and take $\theta=0$ in the proof of Lemma 2.2. We have seen there that $L=\bigoplus_{\pi\in{\cal D}}L(\pi)$ with $L(\pi)\cong O/I_{\pi}$.

Since $\chi^{\prime}_{\pi}$ is now the trivial representation for each $\pi\in{\cal D}$, we have $I_{\pi}\neq 0$ if and only if $\chi_{\lambda,\mskip 1.0mu\mu}(T_{i})=-1$ for at least one $i\in{\cal I}_{\nu(\pi)}$. This condition on $\pi$ means precisely that $\,{\cal I}_{\nu(\pi)}\cap{\cal I}^{1}_{\lambda,\mskip 1.0mu\mu}\neq\mathchar 1343\relax$. Hence

[TABLE]

For each $\sigma\in{{S}}_{n}$ denote by $v_{\sigma}$ the element $T_{\sigma}\otimes 1\in M={\cal H}_{n}(z)\otimes_{{\cal H}_{\rho}(z)}O_{\rm triv}$ and by $v^{\prime}_{\sigma}$ a similar element of $M_{\mskip 1.0mu\mathchar 1404\relax}$. Then $v^{\prime}_{\sigma}$ is the image of $v_{\sigma}$ under the canonical map $M\to M_{\mskip 1.0mu\mathchar 1404\relax}$. Since $L(\pi)$ is the cyclic $O$-submodule of $L$ generated by $\psi(v_{\pi})$, we get

[TABLE]

Put $U_{i}=\{m\in M_{\mskip 1.0mu\mathchar 1404\relax}\mid T_{i}m=-m\}$. Since $(T_{i}+1)^{2}=0$ in the algebra ${\cal H}_{n}(q)$, we have $(T_{i}+1)M_{\mskip 1.0mu\mathchar 1404\relax}\subset U_{i}$. Hence

[TABLE]

If $\pi\in{\cal D}$, then $T_{i}v^{\prime}_{\pi}=-v^{\prime}_{\pi}$, i.e. $v^{\prime}_{\pi}\in U_{i}$, for each $i\in{\cal I}_{\nu(\pi)}$. This shows that $v^{\prime}_{\pi}$ lies in $\Sigma_{\lambda,\mskip 1.0mu\mu}(M_{\mskip 1.0mu\mathchar 1404\relax})$ whenever ${\cal I}_{\nu(\pi)}\cap{\cal I}^{1}_{\lambda,\mskip 1.0mu\mu}\neq\mathchar 1343\relax$. Hence $M_{\mskip 1.0mu\mathchar 1404\relax}^{0}\subset\Sigma_{\lambda,\mskip 1.0mu\mu}(M_{\mskip 1.0mu\mathchar 1404\relax})$.

Conversely, we claim that $U_{i}\subset M_{\mskip 1.0mu\mathchar 1404\relax}^{0}$ for each $i\in{\cal I}^{1}_{\lambda,\mskip 1.0mu\mu}$, which entails the opposite inclusion $\Sigma_{\lambda,\mskip 1.0mu\mu}(M_{\mskip 1.0mu\mathchar 1404\relax})\subset M_{\mskip 1.0mu\mathchar 1404\relax}^{0}$. Fix such an $i$ and consider the Mackey decomposition $M_{\mskip 1.0mu\mathchar 1404\relax}=\bigoplus_{\pi\in{\cal D}}M(\pi)_{\mathchar 1404\relax}$ where $M(\pi)_{\mathchar 1404\relax}$ is the ${\cal H}_{\lambda,\mskip 1.0mu\mu}(q)$-submodule of $M_{\mskip 1.0mu\mathchar 1404\relax}$ generated by $v^{\prime}_{\pi}$. The standard basis of $M(\pi)_{\mathchar 1404\relax}\cong{\cal H}_{\lambda,\mskip 1.0mu\mu}(q)\otimes_{{\cal H}_{\nu(\pi)}(q)}\mathchar 1404\relax_{\rm triv}$ is formed by the elements $v^{\prime}_{\sigma\pi}=T_{\sigma}v^{\prime}_{\pi}$ with $\sigma$ in the set ${\cal D}^{\lambda,\mskip 1.0mu\mu}_{\pi}$ of distinguished representatives of the cosets ${{S}}_{\lambda,\mskip 1.0mu\mu}/{{S}}_{\nu(\pi)}$.

For each $\sigma\in{\cal D}^{\lambda,\mskip 1.0mu\mu}_{\pi}$ either $\tau_{i}\sigma\in{\cal D}^{\lambda,\mskip 1.0mu\mu}_{\pi}$ or, by Deodhar’s Lemma, $\tau_{i}\sigma=\sigma\tau_{j}$ for some $j\in{\cal I}_{\nu(\pi)}$. In the first case $v^{\prime}_{\sigma\pi}$ and $v^{\prime}_{\tau_{i}\sigma\pi}$ span a 2-dimensional $T_{i}$-invariant subspace whose intersection with $U_{i}$ is spanned by a single element

[TABLE]

In the second case $T_{i}T_{\sigma}=T_{\tau_{i}\sigma}=T_{\sigma}T_{j}$ and $T_{j}v^{\prime}_{\pi}=-v^{\prime}_{\pi}$, whence

[TABLE]

Thus $v^{\prime}_{\sigma\pi}$ spans a 1-dimensional $T_{i}$-invariant subspace. Note that $j\in{\cal I}^{1}_{\lambda,\mskip 1.0mu\mu}$ since the transpositions $\tau_{i}$ and $\tau_{j}$ are conjugate in the group ${{S}}_{\lambda,\mskip 1.0mu\mu}$. Thus ${\cal I}_{\nu(\pi)}\cap{\cal I}^{1}_{\lambda,\mskip 1.0mu\mu}\neq\mathchar 1343\relax$, which entails $v^{\prime}_{\pi}\in M_{\mskip 1.0mu\mathchar 1404\relax}^{0}$. But $T_{\sigma}v^{\prime}_{\pi}\equiv(-1)^{\ell(\sigma)}v^{\prime}_{\pi}$ modulo $\mathop{\rm Ker}\psi_{\mskip 1.0mu\mathchar 1404\relax}$ since $\sigma\in{{S}}_{\lambda,\mskip 1.0mu\mu}$ and $(T_{l}+1)M_{\mskip 1.0mu\mathchar 1404\relax}\subset\mathop{\rm Ker}\psi_{\mskip 1.0mu\mathchar 1404\relax}$ for all $l\in{\cal I}_{\lambda,\mskip 1.0mu\mu}$. Therefore $v^{\prime}_{\sigma\pi}\in M_{\mskip 1.0mu\mathchar 1404\relax}^{0}$ too.

The whole $M_{\mskip 1.0mu\mathchar 1404\relax}$ is thus a direct sum of $T_{i}$-invariant subspaces spanned by at most 2 basis elements. It follows that $U_{i}$ is a direct sum of its intersections with those subspaces of $M_{\mskip 1.0mu\mathchar 1404\relax}$. We have checked that $M_{\mskip 1.0mu\mathchar 1404\relax}^{0}$ contains each of the summands in this decomposition of $U_{i}$. Hence $U_{i}\subset M_{\mskip 1.0mu\mathchar 1404\relax}^{0}$, as claimed.   $\mathchar 9219\relax$

Lemma 2.4. The equality \ \sum\limits_{i=0}^{n}(-1)^{i}\,[M_{\mskip 1.0mu\mathchar 1404\relax}/\Sigma_{i,\mskip 1.0mun-i}(M_{\mskip 1.0mu\mathchar 1404\relax})]=0\ holds in the Grothendieck group $\mathop{\rm Grot}\nolimits(\mathop{\rm End}\nolimits_{{\cal H}_{n}(q)}\mskip-2.0muM_{\mathchar 1404\relax})$.

Proof. Consider the parabolic subalgebra ${\cal H}_{i,\mskip 1.0mun-i}(z)_{Q}$ of ${\cal H}_{n}(z)_{Q}$ generated by all elements $T_{j}$ with $0<j<n$, $j\neq i$. Let $Q_{i,\mskip 1.0mun-i}$ be the field $Q$ regarded as the 1-dimensional ${\cal H}_{i,\mskip 1.0mun-i}(z)_{Q}$-module on which $T_{j}$ operates as multiplication by $z$ for $j<i$ and as $\,-\mathop{\rm Id}\nolimits\,$ for $j>i$. By Lemma 2.3

[TABLE]

Since the algebra ${\cal H}_{n}(z)_{Q}$ is semisimple, the functor $?\otimes_{{\cal H}_{n}(z)_{Q}}M_{Q}$ is exact, and therefore this functor induces a group homomorphism

[TABLE]

where we denote by $\mathop{\rm Grot}\nolimits^{r}{\cal H}_{n}(z)_{Q}$ the Grothendieck group of the category of finite dimensional right ${\cal H}_{n}(z)_{Q}$-modules. Lemma 1.1 in its equivalent formulation for right modules shows that

[TABLE]

Applying the above map $g$, we get

[TABLE]

and the desired conclusion follows.   $\mathchar 9219\relax$

Lemma 2.5. There are uniquely determined elements $\zeta_{\lambda}\in\mathop{\rm Grot}\nolimits(\mathop{\rm End}\nolimits_{{\cal H}_{n}(q)}\mskip-2.0muM_{\mathchar 1404\relax})$ with $\lambda\in{\cal P}(n)$ such that

[TABLE]

Moreover,* $\zeta_{\lambda}\geq 0$, i.e. $\zeta_{\lambda}$ represents an actual module*,* for each $\lambda\in{\cal P}(n)$.*

Proof. Let $S^{\lambda}$, $\lambda\in{\cal P}(n)$, be the right Specht modules for the algebra ${\cal H}_{n}(z)_{Q}$, as defined by Dipper and James [6]. By Lemma 1.1

[TABLE]

Applying the group homomorphism $g$ defined in the proof of Lemma 2.4, we get

[TABLE]

Applying now the decomposition map $d$ and making use of Lemma 2.2, we get the desired equalities in the group $\mathop{\rm Grot}\nolimits(\mathop{\rm End}\nolimits_{{\cal H}_{n}(q)}\mskip-2.0muM_{\mathchar 1404\relax})$ with

[TABLE]

We have $\zeta_{\lambda}\geq 0$ since $d$ preserves positivity by the construction. Uniqueness of this collection of elements follows from the fact that the Kostka matrix is invertible. Indeed, this matrix is even unitriangular with respect to a suitable ordering of partitions [21, Ch. I, (6.5)].   $\mathchar 9219\relax$

In the next lemma the assumption that the discrete valuation ring $O$ is complete goes into action.

Lemma 2.6. If $X$ is any direct summand of the ${\cal H}_{n}(q)$-module $M_{\mskip 1.0mu\mathchar 1404\relax},$ then $X=N_{\mskip 1.0mu\mathchar 1404\relax}$ for some direct summand $N$ of the ${\cal H}_{n}(z)$-module $M$.

Proof. There exists an idempotent $e\in\mathop{\rm End}\nolimits_{{\cal H}_{n}(q)}\mskip-2.0muM_{\mathchar 1404\relax}$ such that $X=\mathop{\rm Im}e$. By Lemma 2.1 $\mathop{\rm End}\nolimits_{{\cal H}_{n}(q)}\mskip-2.0muM_{\mathchar 1404\relax}\cong A_{\mskip 0.5mu\mathchar 1404\relax}\cong A/mA$ where $A=\mathop{\rm End}\nolimits_{{\cal H}_{n}(z)}\mskip-2.0muM$ and $m$ is the maximal ideal of $O$. The $O$-algebra $A$ is free of finite rank as a module. It is known that in this situation every idempotent of $A/mA$ lifts to an idempotent of $A$. Hence there exists an idempotent $\mathaccent 869{e}\in A$ such that $e=\mathaccent 869{e}\otimes_{O}\mathchar 1404\relax$. We may take $N=\mathop{\rm Im}\mathaccent 869{e}$.   $\mathchar 9219\relax$

Lemma 2.7. Suppose that $X$ is a finite dimensional ${\cal H}_{n}(q)$-module whose indecomposable direct summands all have $1$-dimensional sources. Then $X=M_{\mskip 1.0mu\mathchar 1404\relax}$ for some $M\in{Rep}^{1}$. If $q=-1,$ then we can even take $M\in{Triv}$.

Proof. The class of ${\cal H}_{n}(q)$-modules for which the conclusion of this lemma holds is obviously closed under direct sums. By Lemma 2.6 this class is closed also under direct summands. Since for $X={\cal H}_{n}(q)\otimes_{{\cal H}_{\lambda,\mu}(q)}\mathchar 1404\relax_{\lambda,\mskip 1.0mu\mu}$ we can take $M=M^{\lambda,\mskip 1.0mu\mu}$, the first assertion holds then in full generality provided $q\neq-1$. If $q=-1$, then any $1$-dimensional representation of a parabolic subalgebra of ${\cal H}_{n}(q)$ is trivial, so that we need only to look at the modules $X={\cal H}_{n}(q)\otimes_{{\cal H}_{\lambda}(q)}\mathchar 1404\relax_{\rm triv}$ for which $M=M^{\lambda}$ will do.   $\mathchar 9219\relax$

Corollary 2.8. If $X$ is as in Lemma 2.7, then:**

(i) \sum\limits_{i=0}^{n}\,(-1)^{i}\,[X/\Sigma_{i,\mskip 1.0mun-i}(X)]=0\ in the Grothendieck group $\,\mathop{\rm Grot}\nolimits(\mathop{\rm End}\nolimits_{{\cal H}_{n}(q)}\mskip-2.0muX),$

*(ii) *there exist finite dimensional $\mathop{\rm End}\nolimits_{{\cal H}_{n}(q)}\mskip-2.0muX$-modules $V^{\lambda},$ $\,\lambda\in{\cal P}(n),$ such that in that group $\,[X/\Sigma_{\mu}(X)]=\sum_{\lambda\in{\cal P}(n)}K_{\lambda\mu}\,[V^{\lambda}]\,$ for each $\mu\in{\cal P}(n)$.

This corollary repeats the conclusions of Lemmas 2.4 and 2.5. Note that

[TABLE]

3. The Hilbert series of the $R$-symmetric algebras

Let $R$ be a Hecke symmetry with parameter $q$ on a finite dimensional vector space $V$ over the ground field $\mathchar 1404\relax$. For each $n\geq 0$ consider the ${\cal H}_{n}(q)$-module structure on $V^{\otimes n}$ arising from $R$. Our first goal in this section is to describe a ring homomorphism $\varphi:\mathop{\rm Sym}\nolimits\to\mathop{\rm Grot}\nolimits(R)$. All essential arguments needed to establish properties of $\varphi$ are provided by the results of section 2.

As a preliminary step we will determine certain quotients of $V^{\otimes n}$. In accordance with the notation introduced in section 2 we have for $(\lambda,\mu)\in{\cal P}^{2}(n)$

[TABLE]

This is an $\mathop{\rm End}\nolimits_{{\cal H}_{n}}\mskip-5.0muV^{\otimes n}$-submodule, i.e. an $A_{n}(R)$-subcomodule, of $V^{\otimes n}$. For each partition $\lambda=(\lambda_{1},\ldots,\lambda_{k})$ put

[TABLE]

It will be assumed that ${{S}}^{\lambda}=\Lambda^{\lambda}=\mathchar 1404\relax$ when $\lambda=0$.

Lemma 3.1. For each $(\lambda,\mu)\in{\cal P}^{2}(n)$ there is an isomorphism of $A_{n}(R)$-comodules

[TABLE]

Proof. Put $l=|\lambda|$ and $m=|\mu|$. Writing $V^{\otimes n}$ as $V^{\otimes l}\otimes V^{\otimes m}$ and noting that $\Sigma_{\lambda,\mskip 1.0mu\mu}(V^{\otimes n})=\Sigma_{\lambda,\mskip 1.0mu0}(V^{\otimes l})\otimes V^{\otimes m}+V^{\otimes l}\otimes\Sigma_{0,\mskip 1.0mu\mu}(V^{\otimes m})$, we get

[TABLE]

The ideal $I^{{S}}$ defining the factor algebra ${{S}}(V,R)$ of the tensor algebra ${{T}}(V)$ has homogeneous components $I^{{S}}_{k}=\sum_{0<i<k}\mathop{\rm Im}(R_{i}^{(k)}-q\cdot\mathop{\rm Id}\nolimits)\subset V^{\otimes k}$. Hence

[TABLE]

and it follows that $V^{\otimes l}/\,\Sigma_{\lambda,\mskip 1.0mu0}(V^{\otimes l})\cong{{S}}^{\lambda}$. On the other hand, the ideal $I^{\Lambda}$ defining $\Lambda(V,R)$ has homogeneous components $I^{\Lambda}_{k}=\sum_{0<i<k}\mathop{\rm Ker}(R_{i}^{(k)}-q\cdot\mathop{\rm Id}\nolimits)$. One obtains similarly $V^{\otimes m}/\,\Sigma_{0,\mskip 1.0mu\mu}(V^{\otimes m})\cong\Lambda^{\mu}$.   $\mathchar 9219\relax$

Proposition 3.2. Suppose that $R$ satisfies the $1$-dimensional source condition. Then there is a ring homomorphism $\varphi:\mathop{\rm Sym}\nolimits\to\mathop{\rm Grot}\nolimits(R)$ such that

(i) $\,\varphi(h_{n})=[{{S}}_{n}(V,R)]$ and $\varphi(e_{n})=[\Lambda_{n}(V,R)]$ for all $n\geq 0,$

(ii) $\,\varphi(s_{\lambda})\geq 0$ for all $\lambda\in{\cal P}$.

Proof. Since $h_{1},h_{2},\ldots$ are algebraically independent generators of the ring of symmetric functions $\mathop{\rm Sym}\nolimits$ (see [21, (2.8)]), homomorphisms from $\mathop{\rm Sym}\nolimits$ to another ring are uniquely determined by their values on those elements. Thus we can define $\varphi$ setting $\varphi(h_{n})=[{{S}}_{n}(V,R)]$ for all $n>0$. Since $h_{0}=e_{0}=1$ and ${{S}}_{0}(V,R)=\Lambda_{0}(V,R)=\mathchar 1404\relax$ is the trivial 1-dimensional $A(R)$-comodule which represents the identity element of the ring $\mathop{\rm Grot}\nolimits(R)$, the two equalities in (i) are obvious for $n=0$. If $n>0$, then the relation $\sum_{i=0}^{n}(-1)^{i}h_{i}e_{n-i}=0$ in $\mathop{\rm Sym}\nolimits$ yields $\sum_{i=0}^{n}(-1)^{i}\varphi(h_{i})\varphi(e_{n-i})=0$. The value $\varphi(e_{n})$ is determined by induction on $n$ from the latter equality. To show that the second equality in (i) holds for all $n>0$ we have to check that

[TABLE]

If $\mu\in{\cal P}(n)$, then $\varphi(h_{\mu})={{S}}{}^{\mskip 0.5mu\mu}$. The equality $h_{\mu}=\sum_{\lambda\in{\cal P}(n)}K_{\lambda\mu}s_{\lambda}$ in $\mathop{\rm Sym}\nolimits$ entails $[\mskip 2.0mu{{S}}{}^{\mskip 0.5mu\mu}]=\sum_{\lambda\in{\cal P}(n)}K_{\lambda\mu}\varphi(s_{\lambda})$. This determines the values of $\varphi$ on the Schur functions since the Kostka matrices are invertible. Part (ii) means that there exists a collection of finite dimensional $A(R)$-comodules $V^{\lambda}$, $\,\lambda\in{\cal P}$, such that $\varphi(s_{\lambda})=[V^{\lambda}]$ for each $\lambda\in{\cal P}$. Those equalities are equivalent to the equalities

[TABLE]

The validity of both (i*′) and (ii′*) in the group $\mathop{\rm Grot}\nolimits_{n}(R)$ is ensured by Corollary 2.8. Indeed, for $X=V^{\otimes n}$ the group $\mathop{\rm Grot}\nolimits\mathop{\rm End}\nolimits_{{\cal H}_{n}(q)}\mskip-2.0muX$ has been identified with $\mathop{\rm Grot}\nolimits_{n}(R)$, and we have

[TABLE]

by Lemma 3.1.   $\mathchar 9219\relax$

Corollary 3.3. In the ring $\,\mathop{\rm Grot}\nolimits(R)[[t]]$ consider the formal power series

[TABLE]

We have $\,G_{{S}}(t)G_{\Lambda}(-t)=1$.

Proof. The coefficient of $t^{n}$ in the product $G_{{S}}(t)G_{\Lambda}(-t)$ vanishes for each $n>0$ according to equality (i*′*) in the proof of Proposition 3.2. The constant term of this product is the identity element $[\mskip 1.0mu\mathchar 1404\relax\mskip 1.0mu]$ of the ring $\mathop{\rm Grot}\nolimits(R)$.   $\mathchar 9219\relax$

Corollary 3.3 strengthens the well known relation $\,H_{{S}}(t)H_{\Lambda}(-t)=1\,$ between the Hilbert series of the algebras ${{S}}={{S}}(V,R)$ and $\Lambda=\Lambda(V,R)$. This relation was proved by Gurevich [15] in the semisimple case. The more general case when $q$ is arbitrary and $R$ satisfies the $1$-dimensional source condition was treated in [24].

For each $\lambda\in{\cal P}$ we will denote by $V^{\lambda}$ any finite dimensional right $A(R)$-comodule such that $[V^{\lambda}]=\varphi(s_{\lambda})$. In the nonsemisimple case such a comodule is generally not unique, but we will be concerned with numeric characteristics such as the dimension of $V^{\lambda}$ which depend only on the class of $V^{\lambda}$ in the group $\mathop{\rm Grot}\nolimits(R)$.

Let $K$ be a commutative ring. For a formal power series $f=\sum_{i=0}^{\infty}a_{i}t^{i}\in K[[t]]$ with $a_{0}=1$ we denote by

[TABLE]

the ring homomorphism such that $h_{n}\mapsto a_{n}$ for each $n\geq 0$. We will assume tacitly that $a_{i}=0$ for all integers $i<0$.

Corollary 3.4. Let $f=\sum_{i=0}^{\infty}a_{i}t^{i}\in{{Z}}[[t]]$ be the Hilbert series of the $R$-symmetric algebra ${{S}}(V,R)$. Then

[TABLE]

for each partition $\lambda$ with $k$ parts $\lambda_{1},\ldots,\lambda_{k}$.

Proof. There is a ring homomorphism $\delta:\mathop{\rm Grot}\nolimits(R)\to{{Z}}$ such that $\delta([X])=\dim X$ for each finite dimensional right $A(R)$-comodule $X$. We have ${\hat{f}}=\delta\circ\varphi$ since the left and right hand sides of this equality are presented by ring homomorphisms with the same values $a_{n}$ on the generators $h_{n}$ of the ring $\mathop{\rm Sym}\nolimits$.

Hence {\hat{f}}(s_{\lambda})=\delta\bigl{(}\varphi(s_{\lambda})\bigr{)}=\dim V^{\lambda}. The second equality follows from the Jacobi-Trudi identity $\,s_{\lambda}=\det\,(h_{\lambda_{i}-i+j})_{1\leq i,j\leq k}\,$.   $\mathchar 9219\relax$

An infinite sequence of real numbers $a_{0},a_{1},a_{2},\ldots$ with $a_{0}=1$ is called totally positive or a Pólya frequency sequence if all minors of the infinite Toeplitz matrix $(a_{j-i})_{i,j\geq 0}$ are nonnegative. Denoting by $f$ the generating series of the given sequence, we see that ${\hat{f}}(s_{\lambda})$ is one of these minors taken from a set of consecutive columns. Other minors are the values of ${\hat{f}}$ on skew Schur functions $s_{\nu/\mu}$. Since $s_{\nu/\mu}=\sum_{\lambda\in{\cal P}}c_{\lambda\mu}^{\nu}s_{\lambda}$ with the Littlewood-Richardson coefficients $c_{\lambda\mu}^{\nu}\geq 0$, it follows that the sequence is totally positive if and only if

[TABLE]

i.e. there is no need to look at the other minors. This formulation of total positivity together with two other equivalent conditions is discussed by Stanley [25, Exercise 7.91e] in the case when $f$ is a polynomial. The statement given above is equally valid for infinite series.

Since $\dim V^{\lambda}\geq 0$ for all $\lambda\in{\cal P}$, it follows from Corollary 3.4 that the dimensions of the homogeneous components ${{S}}_{n}(V,R)$ form a totally positive sequence. All possibilities for the generating series $f$ of a totally positive sequence were determined in the work of Aissen, Schoenberg, Whitney [1] and Edrei [12]. In particular, [1, Th. 1] states that $f$ converges in a neighborhood of [math] in ${{C}}$ and extends to a meromorphic function with negative real zeros and positive real poles on the whole ${{C}}$. As it turns out, $f$ is rational precisely when ${\hat{f}}(s_{\lambda})=0$ for at least one $\lambda\in{\cal P}$.

Thus rationality of the Hilbert series of ${{S}}(V,R)$ is equivalent to the existence of partitions $\lambda$ for which $V^{\lambda}=0$. As observed by Phùng Hô Hai [16], in the semisimple case such $\lambda$ do exist since otherwise the representation of ${\cal H}_{n}$ in $V^{\otimes n}$ would be faithful for each $n$, but this is impossible since $\dim{\cal H}_{n}=n!>(\dim V)^{2n}$ for large $n$. We cannot use this argument directly when $q$ is a root of 1. However, the next lemma provides a replacement.

For integers $a,k>0$ denote by $(a^{k})$ the partition $(a,\ldots,a)$ with exactly $k$ parts, each equal to $a$.

Lemma 3.5. Suppose that $R$ satisfies the $1$-dimensional source condition. There exist integers $n>0$ and $k>0$ such that $V^{(n^{k})}=0$. Moreover,* if $V^{(n^{k})}=0,$ then $V^{(m^{k})}=0$ for all $m>n$.*

Proof. Denote by $d^{\mskip 1.0mu\lambda}$ the dimension of the Specht module $S^{\lambda}$, i.e. $d^{\mskip 1.0mu\lambda}$ is equal to the Kostka number $K_{\lambda,(1^{n})}$ counting the standard $\lambda$-tableaux. We first note that $V^{\lambda}=0$ for each $\lambda\in{\cal P}(n)$ with $d^{\mskip 1.0mu\lambda}>(\dim V)^{n}$. To prove this we start with the equality

[TABLE]

An application of $\varphi$ yields $\,[V^{\otimes n}]=\sum_{\rho\mskip 1.0mu\in{\cal P}(n)}d^{\mskip 1.0mu\rho}\,[V^{\rho}]\,$ in $\mathop{\rm Grot}\nolimits(R)$. Hence

[TABLE]

and therefore $d^{\mskip 1.0mu\lambda}\dim V^{\lambda}\leq(\dim V)^{n}$, which entails the previous claim.

By the hook length formula

[TABLE]

Using the Stirling asymptotic formula $n!\sim n^{n}e^{-n}\sqrt{2\pi n}$ and observing that ${n+i\choose i}\sim n^{i}/i!$ for each $i$, we deduce that

[TABLE]

If $k>\dim V$, then $d^{(n^{k})}>(\dim V)^{kn}$ for large $n$. As we have observed, this yields $V^{(n^{k})}=0$, proving the first assertion of Lemma 3.5.

Next, if $\lambda\in{\cal P}$ is such that $V^{\lambda}=0$, then $0=[V^{\lambda}][V^{\mu}]=\sum_{\nu\in{\cal P}}c_{\lambda\mu}^{\nu}[V^{\nu}]$ for each $\mu\in{\cal P}$. Since $c_{\lambda\mu}^{\nu}\geq 0$ for all $\nu$, it follows that $V^{\nu}=0$ whenever $c_{\lambda\mu}^{\nu}\neq 0$. In particular, this applies in the case when $\lambda=(n^{k})$, $\mu=((m-n)^{k})$ and $\nu=\nobreak(m^{k})$ with $m>n$. Indeed, for these partitions we have $c_{\lambda\mu}^{\nu}=1$ by the Littlewood-Richardson rule.   $\mathchar 9219\relax$

For the application to the Hilbert series of the $R$-symmetric algebras the full strength of the analytic result on totally positive sequences is actually not needed. For one thing rationality of the series follows from a purely algebraic fact formulated in terms of Hankel determinants (see [19, Th. 7.5f]). These determinants with the reversed order of rows are certain minors of the Toeplitz matrix. The next lemma gives a version of that result paying attention to the ring of coefficients.

Lemma 3.6. Let $f=\sum_{i=0}^{\infty}a_{i}t^{i}$ be a formal power series with coefficients in a commutative Noetherian domain $K$. For each pair of integers $i\geq k>0$ put

[TABLE]

Suppose that there are integers $n>r>0$ such that $\Delta_{i}^{(r)}\neq 0$ and $\Delta_{i}^{(r+1)}=0$ for all $i\geq n$. Then $f=q^{-1}p$ for some polynomials $p,q\in K[t]$ with $q(0)=1$. If $K$ is integrally closed,* then such an expression holds with $q$ of degree $r,$ in which case $p$ and $q$ are relatively prime in the ring $Q[t]$ where $Q$ is the field of fractions of $K$.*

Proof. Put $A_{i}^{(k)}=(a_{i-k+1},a_{i-k+2}\,,\ldots,a_{i})$. Part of the hypothesis means that for each $i\geq n$ the $r$ vectors $A_{i}^{(r)}\!,A_{i+1}^{(r)},\ldots,A_{i+r-1}^{(r)}$ are linearly independent over $Q$ and so form a basis for the $r$-dimensional vector space $Q^{r}$. Then

[TABLE]

with uniquely determined coefficients $c_{1},\ldots,c_{r}\in Q$. We claim that these coefficients do not depend on $i$. This will follow once we show that for each $i>n$ the coefficients in $({\rm Rel}_{i})$ are the same as those in $({\rm Rel}_{i-1})$. But $A_{i+r}^{(r+1)}$ is a linear combination of vectors $A_{i}^{(r+1)},A_{i+1}^{(r+1)},\ldots,A_{i+r-1}^{(r+1)}\in Q^{r+1}$ because $\Delta_{i}^{(r+1)}=0$. Since the projection $Q^{r+1}\to Q^{r}$ onto the last $r$ components maps $A_{j}^{(r+1)}$ to $A_{j}^{(r)}$ for each $j$, we must have

[TABLE]

with coefficients from $({\rm Rel}_{i})$. Noting that the projection $Q^{r+1}\to Q^{r}$ onto the first $r$ components maps $A_{j}^{(r+1)}$ to $A_{j-1}^{(r)}$ for each $j$, we get $({\rm Rel}_{i-1})$ with the same coefficients, and the claim is proved.

Now $({\rm Rel}_{i})$ shows that $a_{i+1}=\sum_{j=1}^{r}c_{j}a_{i+1-j}$ for each $i\geq n$. Setting

[TABLE]

we see that the coefficient of $t^{i+1}$ in the formal power series $gf$ vanishes whenever $i\geq n$. Hence $gf$ is a polynomial with coefficients in $Q$.

Define a linear operator $\theta$ on the vector space $Q^{r}$ by the formula

[TABLE]

where $x^{\prime}_{j}=x_{j+1}$ for $0<j<r$ and $x^{\prime}_{r}=\sum_{j=1}^{r}c_{j}x_{r+1-j}$. We have $\theta(A_{i}^{(r)})=A_{i+1}^{(r)}$ for each $i\geq n$, and it follows that

[TABLE]

since this operator annihilates all vectors $A_{i}^{(r)}$ with $i\geq n$ in view of $({\rm Rel}_{i})$. On the other hand, the operators $\theta,\ldots,\theta^{r-1},\theta^{r}$ are linearly independent over $Q$ since so are the vectors $A_{i+1}^{(r)},\ldots,A_{i+r-1}^{(r)},A_{i+r}^{(r)}$ for $i\geq n$. Hence $c_{r}\neq 0$, and $g$ is a scalar multiple of the minimal polynomial of the inverse operator $\theta^{-1}$.

Note that $\theta(M)\subset M$ where $M$ is the submodule of the free $K$-module $K^{r}$ generated by $\{A_{i}^{(r)}\mid i\geq n\}$. Since $K$ is Noetherian, $M$ has to be finitely generated. Hence $\theta$ is integral over $K$, i.e. for some $k>0$ there exists a relation

[TABLE]

with coefficients $e_{j}\in K$. Take $q=1+\sum_{j=1}^{k}e_{j}t^{k}$. Then $g$ divides $q$ since $q(\theta^{-1})=0$, and it follows that $qf$ is a polynomial whose coefficients are in $K$ since so are the coefficients of $f$ and $q$. With $p=qf$ the first conclusion is thus proved.

We can write $q=(1-\xi_{1}t)\cdots(1-\xi_{k}t)$ with $\xi_{1},\ldots,\xi_{k}$ in the algebraic closure of the field $Q$. Each $\xi_{j}$ is integral over $K$ since $q(\xi_{j}^{-1})=0$. Since $g$ is a divisor of $q$ with $g(0)=1$, it is the product of some of these factors $1-\xi_{j}t$. Hence all coefficients of $g$ are integral over $K$ too. If $K$ is integrally closed, we get $g\in K[t]$, and so we may take $q=g$. Then $p$ and $q$ cannot have a common divisor in $Q[t]$ since otherwise $hf\in Q[t]$ for some polynomial $h\in Q[t]$ of degree less than $r$, but this implies that the sequence $(a_{i})$ starting at some term satisfies a linear recurrence relation of order less than $r$, which contradicts the linear independence of the previously considered vectors $A_{i}^{(r)}\!,A_{i+1}^{(r)},\ldots,A_{i+r-1}^{(r)}$.   $\mathchar 9219\relax$

Knowing rationality of $f$, one needs only part of the arguments given in [1] to determine the location of zeros and poles. Moreover, Stanley’s criterion of total positivity provides further simplifications:

Lemma 3.7. Let $f=\sum_{i=0}^{\infty}a_{i}t^{i}\in{{R}}[[t]]$ be the generating series of a totally positive sequence. Suppose that $f$ represents a rational function of $t$. Then all its zeroes are negative,* while all its poles are positive real numbers.*

Proof. If $a_{n}\neq 0$ and $a_{n+1}=0$ for some $n\geq 0$, then $f$ is a polynomial since for each $i>n$ the conditions $a_{i}a_{n+1}-a_{i+1}a_{n}\geq 0$ and $a_{i+1}a_{n}\geq 0$ entail $a_{i+1}=0$.

Suppose that $a_{n}\neq 0$ for all $n\geq 0$. Then the positive numbers $a_{n+1}/a_{n}$ form a monotone nonincreasing sequence. Hence this sequence converges to some $\gamma\geq 0$. Rationality of $f$ implies that $\gamma^{-1}$ is one of its poles, i.e. $1-\gamma t$ is a divisor of the denominator $q$ in the expression of $f$ as a fraction of two relatively prime polynomials. In particular, $\gamma\neq 0$. The power series $f_{\flat}=(1-\gamma t)f$ has coefficients

[TABLE]

which form a totally positive sequence. Indeed, given a partition $\lambda=(\lambda_{1},\ldots,\lambda_{k})$, we have

[TABLE]

Since $\,\gamma^{i}=\lim_{n\to\infty}a_{n+i}/a_{n}\,$, this yields

[TABLE]

where we put $\lambda^{(n)}=(n,\lambda_{1},\ldots,\lambda_{k})$. Thus $f_{\flat}$ satisfies the same assumptions as $f$. We have seen that $f$ has a pole $\gamma^{-1}>0$. Proceeding by induction on the degree of the denominator $q$, we conclude that all poles of $f$ are positive.

The coefficients $b_{i}$ of the power series $g(t)=1/f(-t)$ also form a totally positive sequence. This fact was proved in [1], but again it can be explained very easily within the theory of symmetric functions. Since $\sum_{i=0}^{n}(-1)^{i}a_{i}b_{n-i}=0$ for $n>0$, we have $b_{n}={\hat{f}}(e_{n})$. Hence ${\hat{g}}={\hat{f}}\circ\omega$ where $\omega$ is the automorphism of the ring $\mathop{\rm Sym}\nolimits$ such that $h_{n}\mapsto e_{n}$ for each $n$. Since $\omega(s_{\lambda})=s_{\lambda^{\prime}}$ where $\lambda^{\prime}$ is the conjugate of $\lambda$, we get ${\hat{g}}(s_{\lambda})={\hat{f}}(s_{\lambda^{\prime}})\geq 0$ for each $\lambda\in{\cal P}$. Thus all poles of $g$ are positive, which means that all zeroes of $f$ are negative.   $\mathchar 9219\relax$

Theorem 3.8. Suppose that $R$ satisfies the $1$-dimensional source condition. Then

[TABLE]

with integer polynomials $\,f_{0},\,f_{1}\in{{Z}}[t]\,$ whose constant terms are equal to $1$ and all roots are positive real numbers.

Proof. Put $f=H_{{{S}}(V,R)}$. Since $H_{\Lambda(V,R)}(t)=f(-t)^{-1}$, it suffices to prove only the formula for $f$. Define $\Delta_{i}^{(k)}$ as in Lemma 3.6. By Corollary 3.4 $f$ is the generating series of a totally positive sequence, and also $\Delta_{i}^{(k)}=\dim V^{(i^{k})}$. Let $r$ be the smallest nonnegative integer for which there exists $n>0$ such that $V^{(n^{r+1})}=0$. By Lemma 3.5 $r$ is well-defined and $V^{(m^{r+1})}=0$ for all $m>n$.

If $r=0$, then ${{S}}_{n}(V,R)=0$, which means that $f$ is a polynomial. Note that the constant term of $f$ is equal to 1.

If $r>0$, then the determinants $\Delta_{i}^{(k)}$ satisfy the assumption of Lemma 3.6. Taking $K={{Z}}$, we deduce that $f$ is a fraction of two relatively prime integer polynomials with constant terms equal to 1. By Lemma 3.7 $f$ has negative zeros and positive poles. This ensures the desired properties of $f_{0}$ and $f_{1}$.   $\mathchar 9219\relax$

Now we will extend to the present situation two additional results obtained by Phùng Hô Hai in the semisimple case [16, Th. 5.1, Cor. 5.2].

Corollary 3.9. Let $(r_{0},r_{1})$ be the birank of $R,$ i.e. $r_{i}=\mathop{\rm deg}\nolimits f_{i}$ for $i=0,1$. Then $r_{0}+r_{1}\leq\dim V$. Moreover,* if $r_{0}+r_{1}=\dim V,$ then*

[TABLE]

Proof. We have $f_{0}(t)=\prod_{i=1}^{r_{0}}(1-\alpha_{i}t)$ and $f_{1}(t)=\prod_{i=1}^{r_{1}}(1-\beta_{i}t)$ with $\alpha_{i},\beta_{i}>0$. The fact that all coefficients of these polynomials are integers entails

[TABLE]

Since the coefficient of $t$ in $H_{{{S}}(V,R)}$ is equal to the dimension of $V$, we get

[TABLE]

If the equality is attained here, then $\sum\alpha_{i}=r_{0}$ and $\sum\beta_{j}=r_{1}$, so that the equalities are attained also in the previously displayed formulas. This is only possible when all $\alpha_{i}$ and $\beta_{j}$ are equal to 1.   $\mathchar 9219\relax$

Corollary 3.10. Let $(r_{0},r_{1})$ be the birank of $R$. Then $V^{\lambda}\neq 0$ if and only if $\lambda_{j}\leq r_{1}$ for all $j>r_{0},$ i.e. $\lambda\in\Gamma(r_{0},r_{1})$.

Proof. Let $x_{1},\ldots,x_{r_{0}},y_{1},\ldots,y_{r_{1}}$ be commuting indeterminates. Consider the ring homomorphism $\mathop{\rm Sym}\nolimits\to{{Z}}[x_{1},\ldots,x_{r_{0}},y_{1},\ldots,y_{r_{1}}]$ under which the formal power series $\sum_{n=0}^{\infty}e_{n}t^{n}$ and $\sum_{n=0}^{\infty}h_{n}t^{n}$ specialize, respectively, to

[TABLE]

Denote by $u(x/y)$ the image of $u\in\mathop{\rm Sym}\nolimits$ in the ring ${{Z}}[x_{1},\ldots,x_{r_{0}},y_{1},\ldots,y_{r_{1}}]$ and by $u(\alpha/\beta)$ the value of this polynomial $u(x/y)$ at the point

[TABLE]

where $\alpha_{i}$ and $\beta_{j}$ are as in the proof of Corollary 3.9. Then

[TABLE]

Since the left and right hand sides of this equality are evaluations at $s_{\lambda}$ of two ring homomorphisms $\mathop{\rm Sym}\nolimits\to{{Z}}$, it suffices to check it on the generators $s_{(n)}=h_{n}$ of the ring $\mathop{\rm Sym}\nolimits$. But for $\lambda=(n)$ we have $V^{\lambda}={{S}}_{n}(V,R)$, while $s_{\lambda}(\alpha/\beta)=h_{n}(\alpha/\beta)$ is exactly the coefficient of $t^{n}$ in the Hilbert series of the algebra ${{S}}(V,R)$, so that the equality is indeed true.

The specialization $u\mapsto u(x/y)$ defined above differs from that of Macdonald [21, Ch. I, Example 3.23] in that each $y_{j}$ is changed to $-y_{j}$. With this change formula (1) in [21, Ch. I, Example 5.23] reads as

[TABLE]

where $s_{\lambda^{\prime}\!/\mu^{\prime}}$ is the skew Schur function corresponding to the pair $\lambda^{\prime},\mu^{\prime}$ of partitions conjugate to $\lambda$ and $\mu$. Thus $s_{\lambda}(x/y)$ is precisely the hook Schur function $HS_{\lambda}$ in the notation and terminology of Berele and Regev [3, Definition 6.3].

Recall that each skew Schur function is a linear combination of monomial symmetric functions with nonnegative integer coefficients. Hence any monomial in the indeterminates $x_{1},\ldots,x_{r_{0}},y_{1},\ldots,y_{r_{1}}$ has nonnegative coefficient in $s_{\lambda}(x/y)$. It is known also that $s_{\lambda}(x/y)\neq 0$ if and only if $\lambda\in\Gamma(r_{0},r_{1})$ [3, Cor. 6.5]. Since all real numbers $\alpha_{i}$ and $\beta_{j}$ are positive, we deduce that $\,\dim V^{\lambda}=s_{\lambda}(\alpha/\beta)>0\,$ if and only if $\lambda\in\Gamma(r_{0},r_{1})$.   $\mathchar 9219\relax$

**Remark. **By Corollary 3.10 the image of $\varphi$ is the subgroup of $\mathop{\rm Grot}\nolimits(R)$ generated by the classes $[V^{\lambda}]$ with $\lambda\in\Gamma(r_{0},r_{1})$. It is not clear whether these classes are always linearly independent over ${{Z}}$. In any event for each $n\geq 0$ the rank of the free abelian group $\varphi(\mathop{\rm Sym}\nolimits_{n})$ does not exceed the cardinality of the set $\Gamma(r_{0},r_{1})\cap{\cal P}(n)$. On the other hand, the group $\mathop{\rm Grot}\nolimits_{n}(R)$ may have a larger rank. When this happens, $\varphi$ is not surjective unlike what we have seen in the semisimple case.

As an example consider the supersymmetry $R$ on a ${{Z}}/2{{Z}}$-graded vector space $V=V_{0}\oplus V_{1}$. It is defined by the rule $R(v\otimes w)=w\otimes v$ for homogeneous elements $v,w\in V$ at least one of which is even, and $R(v\otimes w)=-w\otimes v$ when both $v$ and $w$ are odd. We assume here that $\mathop{\rm char}\nolimits\mathchar 1404\relax\neq 2$. This operator is a Hecke symmetry with parameter $q=1$, so that ${\cal H}_{n}(q)$ is the group algebra $\mathchar 1404\relax{{S}}_{n}$. It is easy to see that the $1$-dimensional source condition is satisfied. Since ${{S}}(V,R)$ is the tensor product $S(V_{0})\otimes\bigwedge(V_{1})$ of the ordinary symmetric and exterior algebras, it has Hilbert series $(1+t)^{r_{1}}/(1-t)^{r_{0}}$ where $r_{i}=\dim V_{i}$ for $i=0,1$. Thus the birank of $R$ coincides with the superdimension of $V$.

The right $A_{n}(R)$-comodules may be identified with the degree $n$ polynomial representations of the supergroup $GL_{r_{0}\mid r_{1}}$ over the field $\mathchar 1404\relax$. (In the framework of super theory one endows $A(R)=\bigoplus_{k=0}^{\infty}A_{k}(R)$ with a modified version of the multiplication described in section 1. This modified multiplication makes $A(R)$ into a super bialgebra rather than an ordinary bialgebra, and in this way $A(R)$ is identified with the subalgebra of the coordinate algebra of $GL_{r_{0}\mid r_{1}}$ generated by the coefficient functions of the natural representation on $V$.)

Thus the rank of $\mathop{\rm Grot}\nolimits_{n}(R)$ equals the number of irreducible polynomial representations of $GL_{r_{0}\mid r_{1}}$ of degree $n$. Irreducible representations are classified by their highest weights with respect to a maximal torus $T$ of the group $GL_{r_{0}}\times GL_{r_{1}}\subset GL_{r_{0}\mid r_{1}}$. Highest weights of polynomial representations may be interpreted as pairs of partitions $\lambda,\mu$ such that $\ell(\lambda)\leq r_{0}$ and $\ell(\mu)\leq r_{1}$. However, not all such pairs correspond to a polynomial representation.

If $\mathop{\rm char}\nolimits\mathchar 1404\relax=0$, then the highest weights of irreducible polynomial representations of $GL_{r_{0}\mid r_{1}}$ are selected by the additional condition $\ell(\mu)\leq\lambda_{r_{0}}$ on the pair $(\lambda,\mu)$ (see [23, Cor. 1 to Th. 2]). On the other hand, since the group algebras $\mathchar 1404\relax{{S}}_{n}$ are semisimple, the irreducible polynomial $GL_{r_{0}\mid r_{1}}$-modules of degree $n$ are precisely the simple $A_{n}(R)$-comodules

[TABLE]

with $\nu\in\Gamma(r_{0},r_{1})\cap{\cal P}(n)$, while $V^{\nu}_{R}=0$ when $\nu\notin\Gamma(r_{0},r_{1})$. In terms of representations of Lie superalgebras this fact was established long ago independently by Sergeev [23] and Berele, Regev [3]. It should be noted also that the character of $V^{\nu}_{R}$ defined in terms of weight spaces with respect to the torus $T$ is exactly the hook Schur function $s_{\nu}(x/y)$. This explains the properties of these functions referred to in the proof of Corollary 3.10.

Suppose now that $\mathop{\rm char}\nolimits\mathchar 1404\relax=p>0$. In this case the irreducible representation of $GL_{r_{0}\mid r_{1}}$ with highest weight represented by the pair $(\lambda,\mu)$ is polynomial if and only if $j(\mu)\leq\lambda_{r_{0}}$ where $j(\mu)$ is the cardinality of a combinatorially defined subset of nodes of the Young diagram of $\mu$ which contains at most one node from each row of the diagram. This condition was found by Brundan and Kujawa [4, Th. 6.5]. The inequality $j(\mu)\leq\ell(\mu)$ holds for each $\mu$. At the same time there exist highest weights $(\lambda,\mu)$ such that $\ell(\mu)>\lambda_{r_{0}}$ but $j(\mu)\leq\lambda_{r_{0}}$. For example, $j(\mu)=0$ if $\mu_{i}\equiv 0\mskip 12.0mu(\mathop{\rm mod}p)$ for all $i$. From this it follows that the group $\mathop{\rm Grot}\nolimits_{n}(R)$ has larger rank than that in the case of a field of characteristic 0, and therefore $\varphi(\mathop{\rm Sym}\nolimits_{n})\neq\mathop{\rm Grot}\nolimits_{n}(R)$, for infinitely many $n$.

4. Tensor powers $V^{\otimes n}$ as modules over the Hecke algebras

Consider $V^{\otimes n}$ as an ${\cal H}_{n}(q)$-module with respect to the representation arising from a Hecke symmetry $R$. Assuming that $R$ satisfies the $1$-dimensional source condition, we will associate with this module a symmetric function $\mathop{\rm ch}\nolimits([V^{\otimes n}])\in\mathop{\rm Sym}\nolimits_{n}$ which encodes enough information to determine the image of $V^{\otimes n}$ in the Grothendieck group $\mathop{\rm Grot}\nolimits{\cal H}_{n}(q)$.

For an associative algebra ${{A}}$ over some field, say $F$, denote by $\,\mathop{\rm Rep}{{A}}\,$ the abelian group generated by the isomorphism classes $[X]$ of finite dimensional left ${{A}}$-modules with the defining relations $[X]=[X^{\prime}]+[X^{\prime\prime}]$ for each triple of finite dimensional left ${{A}}$-modules such that $X\cong X^{\prime}\oplus X^{\prime\prime}$. It is a free abelian group with a basis consisting of the isomorphism classes of indecomposable finite dimensional left ${{A}}$-modules. By abuse of notation $[X]$ will stand for an element of either $\mathop{\rm Rep}{{A}}$ or $\mathop{\rm Grot}\nolimits{{A}}$, depending on the context. There is a canonical group homomorphism

[TABLE]

sending the class of $X$ in $\mathop{\rm Rep}{{A}}$ to the class of $X$ in $\mathop{\rm Grot}\nolimits{{A}}$. This map is an isomorphism when ${{A}}$ is semisimple. In general $X\cong Y$ whenever $[X]=[Y]$ in $\mathop{\rm Rep}{{A}}$.

Consider a ${{Z}}$-bilinear form on $\mathop{\rm Rep}{{A}}$ defined by the formula

[TABLE]

for each pair of finite dimensional left ${{A}}$-modules $X$ and $Y$.

Denote by ${\rm Rep}^{1}\mskip 1.0mu{\cal H}_{n}(q)$ (respectively, $\mathop{\rm Triv}{\cal H}_{n}(q)$) the subgroup of $\mathop{\rm Rep}{\cal H}_{n}(q)$ generated by the isomorphism classes of indecomposable left ${\cal H}_{n}(q)$-modules which have a 1-dimensional (respectively, trivial) source. Then $\,\mathop{\rm Triv}{\cal H}_{n}(q)\subset{\rm Rep}^{1}\mskip 1.0mu{\cal H}_{n}(q)$.

In the next lemma we work in the settings of section 2:

Lemma 4.1. There is a group homomorphism $\,e:{\rm Rep}^{1}\mskip 1.0mu{\cal H}_{n}(q)\to\mathop{\rm Grot}\nolimits{\cal H}_{n}(z)_{Q}\,$ such that $\,e([M_{\mskip 1.0mu\mathchar 1404\relax}])=[M_{Q}]\,$ for each lattice $M\in{Rep}^{1}$ if $q\neq-1$ and for $M\in{Triv}$ if $q=-1$. It makes commutative the diagram

[TABLE]

where $d$ is the decomposition map and $c$ is the canonical map.

Proof. Since ${\cal H}_{n}(z)_{Q}$ is semisimple, the classes of Specht modules $S^{\lambda}_{Q}$, $\lambda\in{\cal P}(n)$, for this algebra form a ${{Z}}$-basis of $\mathop{\rm Grot}\nolimits{\cal H}_{n}(z)_{Q}$. Since these modules are absolutely irreducible, we have $\,\langle S^{\lambda}_{Q},S^{\mu}_{Q}\rangle=\delta_{\lambda\mu}$ (the Kronecker symbol). In particular the bilinear form on $\mathop{\rm Grot}\nolimits{\cal H}_{n}(z)_{Q}$ is nondegenerate. The equality of dimensions in Lemma 2.1 can be restated by saying that

[TABLE]

for any two lattices $M,N\in{Rep}^{1}$ if $q\neq-1$ and for $M,N\in{Triv}$ if $q=-1$.

If $M,M^{\prime}\in{Rep}^{1}$ are such that $M_{\mskip 1.0mu\mathchar 1404\relax}\cong M_{\mskip 1.0mu\mathchar 1404\relax}^{\prime}$ as ${\cal H}_{n}(q)$-modules, and if moreover $M,M^{\prime}\in{Triv}$ when $q=-1$, then it follows from the displayed equality that $\langle X,M_{Q}\rangle=\langle X,M^{\prime}_{Q}\rangle$ for each permutation ${\cal H}_{n}(z)_{Q}$-module $X$, i.e. a module induced from the trivial 1-dimensional representation of a parabolic subalgebra of ${\cal H}_{n}(z)_{Q}$. Since the classes of permutation modules form a ${{Z}}$-basis of $\mathop{\rm Grot}\nolimits{\cal H}_{n}(z)_{Q}$, we conclude that $M_{Q}\cong M^{\prime}_{Q}$ in this case.

This shows that the map $e$ is well-defined on the elements $[M_{\mathchar 1404\relax}]$. By Lemma 2.7 each indecomposable ${\cal H}_{n}(q)$-module with a 1-dimensional source is isomorphic to $M_{\mathchar 1404\relax}$ for a suitable choice of $M$. Hence $e$ is well-defined on the semigroup of positive elements in ${\rm Rep}^{1}\mskip 1.0mu{\cal H}_{n}(q)$. Since both $?\otimes_{O}\mathchar 1404\relax$ and $?\otimes_{O}Q$ are additive functors, we have $e(a+b)=e(a)+e(b)$ for any pair of positive elements $a,b\in{\rm Rep}^{1}\mskip 1.0mu{\cal H}_{n}(q)$. It follows that $e$ extends to a group homomorphism on the whole ${\rm Rep}^{1}\mskip 1.0mu{\cal H}_{n}(q)$. Commutativity of the diagram is clear from the definition of $d$ in section 2.   $\mathchar 9219\relax$

Proposition 4.2. There exist group homomorphisms $\mathop{\rm ch}\nolimits:{\rm Rep}^{1}\mskip 1.0mu{\cal H}_{n}(q)\to\mathop{\rm Sym}\nolimits_{n}$ and $\psi:\mathop{\rm Sym}\nolimits_{n}\to\mathop{\rm Grot}\nolimits{\cal H}_{n}(q)$ with the following properties:**

(i) $\psi\circ\mathop{\rm ch}\nolimits$ is equal to the canonical map $c:{\rm Rep}^{1}\mskip 1.0mu{\cal H}_{n}(q)\to\mathop{\rm Grot}\nolimits{\cal H}_{n}(q),$

(ii) $\psi(h_{\lambda}e_{\mu})=[{\cal H}_{n}(q)\otimes_{{\cal H}_{{\lambda,\mskip 1.0mu\mu}}(q)}\mathchar 1404\relax_{\lambda,\mskip 1.0mu\mu}]$ for each pair $({\lambda,\mskip 1.0mu\mu})\in{\cal P}^{2}(n),$

*(iii) *if $q\neq-1$ then $\mathop{\rm ch}\nolimits([{\cal H}_{n}(q)\otimes_{{\cal H}_{{\lambda,\mskip 1.0mu\mu}}(q)}\mathchar 1404\relax_{\lambda,\mskip 1.0mu\mu}])=h_{\lambda}e_{\mu}\,$ for each pair $({\lambda,\mskip 1.0mu\mu})\in{\cal P}^{2}(n),$

(iv) $\mathop{\rm ch}\nolimits([{\cal H}_{n}(q)\otimes_{{\cal H}_{\lambda}(q)}\mathchar 1404\relax_{\rm triv}])=h_{\lambda}$ for each $\lambda\in{\cal P}(n),$

(v) $\langle\mathop{\rm ch}\nolimits(a),\mathop{\rm ch}\nolimits(b)\,\rangle=\langle a,b\rangle$ for all $\,a,b\in{\rm Rep}^{1}\mskip 1.0mu{\cal H}_{n}(q)$.

(vi) $\mathop{\rm ch}\nolimits$ maps the subgroup $\,\mathop{\rm Triv}{\cal H}_{n}(q)\,$ isomorphically onto $\,\mathop{\rm Sym}\nolimits_{n}$,

Proof. By the semisimple case recalled in section 1 there is a canonical isomorphism

[TABLE]

Composing the group homomorphisms $d$ and $e$ of Lemma 4.1 with the previous isomorphism, we obtain $\psi$ and $\mathop{\rm ch}\nolimits$, respectively. The commutative diagram in Lemma 4.1 yields (i).

Now take $M=M^{\lambda,\mskip 1.0mu\mu}$, i.e. $M={\cal H}_{n}(z)\otimes_{{\cal H}_{\lambda,\mu}(z)}O_{\lambda,\mskip 1.0mu\mu}\in{Rep}^{1}$ (see section 2). We have $d([M_{Q}])=[M_{\mskip 1.0mu\mathchar 1404\relax}]$ by the definition of $d$. If $q\neq-1$ then $e([M_{\mskip 1.0mu\mathchar 1404\relax}])=[M_{Q}]$ by the definition of $e$. If $\mu=0$, so that $M=M^{\lambda}\in{Triv}$, then $e([M_{\mskip 1.0mu\mathchar 1404\relax}])=[M_{Q}]$ for any $q$. Since

[TABLE]

corresponds to $h_{\lambda}e_{\mu}\in\mathop{\rm Sym}\nolimits_{n}$ by Lemma 1.1, we get (ii)–(iv).

By Lemma 2.7 the group ${\rm Rep}^{1}\mskip 1.0mu{\cal H}_{n}(q)$ is generated by the classes $[N_{\mskip 1.0mu\mathchar 1404\relax}]$ for all lattices $N\in{Rep}^{1}$, and when $q=-1$ it suffices to use only the lattices $N\in{Triv}$. Since $e([N_{\mskip 1.0mu\mathchar 1404\relax}])=[N_{Q}]$ for such lattices, it follows from Lemma 2.1 that

[TABLE]

for all $a,b\in{\rm Rep}^{1}\mskip 1.0mu{\cal H}_{n}(q)$. Since the isomorphism $\mathop{\rm Sym}\nolimits_{n}\cong\mathop{\rm Grot}\nolimits{\cal H}_{n}(z)_{Q}$ is isometric with respect to the scalar products defined on these groups, we get (v).

The trivial source indecomposable ${\cal H}_{n}(q)$-modules are precisely the Young modules $Y^{\lambda}$ parametrized by partitions $\lambda\in{\cal P}(n)$. These modules were described by Dipper and James in [7, Lemma 2.5]. For each $\lambda$ the “permutation” module

[TABLE]

has $Y^{\lambda}$ as its direct summand of multiplicity 1. All other direct summands of this permutation module are the Young modules $Y^{\nu}$ with $\nu>\lambda$ with respect to the dominance order on partitions.

The isomorphism classes of Young modules $Y^{\lambda}$ form a ${{Z}}$-basis of $\mathop{\rm Triv}{\cal H}_{n}(q)$. It follows that the classes of permutation modules also form a ${{Z}}$-basis of $\mathop{\rm Triv}{\cal H}_{n}(q)$. According to (iv) $\mathop{\rm ch}\nolimits$ maps this basis of $\mathop{\rm Triv}{\cal H}_{n}(q)$ to the ${{Z}}$-basis $\{h_{\lambda}\mid\lambda\in{\cal P}(n)\}$ of $\mathop{\rm Sym}\nolimits_{n}$. This entails (vi).   $\mathchar 9219\relax$

**Remark. **Actually the fact that the permutation ${\cal H}_{n}(q)$-modules do not have any indecomposable direct summands other than the Young modules was not proved in [7] for arbitrary $q$. The isomorphism classes of those indecomposable summands are in a bijection with the isomorphism classes of simple modules for the $q$-Schur algebra $S_{q}(k,n)$ when $k\geq n$. In a later paper Dipper and James showed that the latter modules are parametrized by partitions of $n$ [8, Th. 8.8]. This settled the question about the direct summands of permutation modules (see Donkin [9, 4.4]).

Let us return to consideration of the Hecke symmetry $R$. Put ${\cal H}_{n}={\cal H}_{n}(q)$.

Lemma 4.3. Let $f$ be the Hilbert series of the $R$-symmetric algebra ${{S}}(V,R)$. Then

[TABLE]

for each $\nu\in{\cal P}(n)$.

Proof. By the Frobenius reciprocity

[TABLE]

where $\Sigma_{\nu}(V^{\otimes n})=\sum_{i\in{\cal I}_{\nu}}(T_{i}-q)V^{\otimes n}$. If $\nu=(\nu_{1},\ldots,\nu_{k})$, then

[TABLE]

by Lemma 3.1. Since $\,\dim{{S}}_{\nu_{i}}(V,R)={\hat{f}}(h_{\nu_{i}})$, we get

[TABLE]

which yields the first equality in the statement of Lemma 4.3. The second equality follows from the fact that the two bases $\{m_{\lambda}\mid\lambda\in{\cal P}(n)\}$ and $\{h_{\lambda}\mid\lambda\in{\cal P}(n)\}$ of the group $\mathop{\rm Sym}\nolimits_{n}$ are dual to each other with respect to the scalar product on $\mathop{\rm Sym}\nolimits_{n}$ [21, Ch. I, (4.5)]. Hence $u=\sum_{\lambda\in{\cal P}(n)}\langle h_{\lambda},u\rangle m_{\lambda}$ for each $u\in\mathop{\rm Sym}\nolimits_{n}$, and so this formula can be used for $u=h_{\nu}$.   $\mathchar 9219\relax$

Lemma 4.4. If $f=\prod\limits_{j=1}^{r_{1}}(1+\beta_{j}t)\cdot\prod\limits_{i=1}^{r_{0}}(1-\alpha_{i}t)^{-1},$ then

[TABLE]

where we put $u(\alpha)=u(\alpha_{1},\ldots,\alpha_{r_{0}}),$ $u(\beta)=u(\beta_{1},\ldots,\beta_{r_{1}})$ for each $u\in\mathop{\rm Sym}\nolimits$ and $N_{({\lambda,\mskip 1.0mu\mu}),\mskip 1.0mu\nu}$ is the number of all pairs of nonnegative integer matrices

[TABLE]

such that $B$ has only entries equal to [math] or $1,$

[TABLE]

Also,* $\,N_{({\lambda,\mskip 1.0mu\mu}),\mskip 1.0mu\nu}=\langle h_{\lambda}e_{\mu},h_{\nu}\rangle$.*

Proof. Recall that $\,\prod(1-\alpha_{i}t)^{-1}=\sum h_{p}(\alpha)t^{p}\,$ and $\,\prod(1+\beta_{j}t)=\sum e_{p}(\beta)t^{p}$ by the definitions of the complete and elementary symmetric functions. For each integer $p\geq 0$ the coefficient of $t^{p}$ in $f$ is

[TABLE]

i.e. ${\hat{f}}(h_{p})$ equals the sum of all monomials $\alpha_{1}^{a_{1}}\cdots\alpha_{r_{0}}^{a_{r_{0}}}\beta_{1}^{b_{1}}\cdots\beta_{r_{1}}^{b_{r_{1}}}$ with integer exponents $a_{i}\geq 0$ and $b_{j}\in\{0,1\}$ such that $\sum a_{i}+\sum b_{j}=p$. Hence

[TABLE]

where $k=\ell(\nu)$ and the sum runs over all pairs of nonnegative integer matrices

[TABLE]

such that $B$ has only entries equal to 0 or 1 and

[TABLE]

For $\lambda,\mu\in{\cal P}$ we have $m_{\lambda}(\alpha)=0$ whenever $\ell(\lambda)>r_{0}$ and $m_{\mu}(\beta)=0$ whenever $\ell(\mu)>r_{1}$. If $\ell(\lambda)\leq r_{0}$ and $\ell(\mu)\leq r_{1}$, then each occurrence of the monomial $\alpha_{1}^{\lambda_{1}}\cdots\alpha_{r_{0}}^{\lambda_{r_{0}}}\beta_{1}^{\mu_{1}}\cdots\beta_{r_{1}}^{\mu_{r_{1}}}$ in the previous expression for ${\hat{f}}(h_{\nu})$ corresponds to a pair of matrices $A,B$ used in the definition of $N_{({\lambda,\mskip 1.0mu\mu}),\mskip 1.0mu\nu}$. Thus $N_{({\lambda,\mskip 1.0mu\mu}),\mskip 1.0mu\nu}$ is the total number of such occurrences. Since ${\hat{f}}(h_{\nu})$ is a symmetric function of $\alpha_{1},\ldots,\alpha_{r_{0}}$ and a symmetric function of $\beta_{1},\ldots,\beta_{r_{1}}$, it is a ${{Z}}$-linear combination of the products of monomial symmetric functions in these sets of elements, and we obtain the desired formula.

Next, writing out the product $h_{\lambda}e_{\mu}=h_{\lambda_{1}}\cdots h_{\lambda_{\ell(\lambda)}}e_{\mu_{1}}\cdots e_{\mu_{\ell(\mu)}}$ as a sum of monomials in the indeterminates $x_{1},x_{2},\ldots\,$, we see that the coefficient of $x_{1}^{\nu_{1}}\cdots x_{k}^{\nu_{k}}$ in $h_{\lambda}e_{\mu}$ equals $N_{({\lambda,\mskip 1.0mu\mu}),\mskip 1.0mu\nu}$ too. Hence

[TABLE]

and it follows that $\,\langle h_{\lambda}e_{\mu},h_{\nu}\rangle=N_{({\lambda,\mskip 1.0mu\mu}),\mskip 1.0mu\nu}\,$.   $\mathchar 9219\relax$

We will write $\,\mathop{\rm ch}\nolimits(X)=\mathop{\rm ch}\nolimits([X])\,$ for each finite dimensional left ${\cal H}_{n}$-module $X$ where $\,\mathop{\rm ch}\nolimits\,$ is the map of Proposition 4.2. The property

[TABLE]

of this map enables us to determine $\mathop{\rm ch}\nolimits(V^{\otimes n})$:

Theorem 4.5. Let $R:V^{\otimes 2}\to V^{\otimes 2}$ be a Hecke symmetry of birank $(r_{0},r_{1})$ with parameter $q$. Suppose that $R$ satisfies the $1$-dimensional source condition and

[TABLE]

Then  $\displaystyle\mathop{\rm ch}\nolimits(V^{\otimes n})=\sum_{({\lambda,\mskip 1.0mu\mu})\in{\cal P}^{2}(n)}\,m_{\lambda}(\alpha)m_{\mu}(\beta)h_{\lambda}e_{\mu}\,$. In particular,**

[TABLE]

in the Grothendieck group $\mathop{\rm Grot}\nolimits{\cal H}_{n}(q)$.

Proof. By Lemmas 4.3 and 4.4

[TABLE]

for each $\nu\in{\cal P}(n)$. Also, $\,\langle\mathop{\rm ch}\nolimits(V^{\otimes n}),h_{\nu}\rangle=\langle V^{\otimes n},{\cal H}_{n}\otimes_{{\cal H}_{\nu}}\mathchar 1404\relax_{\rm triv}\rangle\,$ by (iv) and (v) of Proposition 4.2. This means that

[TABLE]

is orthogonal to all functions $h_{\nu}$, $\nu\in{\cal P}(n)$, which generate the whole group $\mathop{\rm Sym}\nolimits_{n}$. Nondegeneracy of the scalar product entails the required formula for $\mathop{\rm ch}\nolimits(V^{\otimes n})$. The final equality is obtained then by applying the map $\psi:\mathop{\rm Sym}\nolimits_{n}\to\mathop{\rm Grot}\nolimits{\cal H}_{n}$ described in Proposition 4.2.   $\mathchar 9219\relax$

Corollary 4.6. If $n$ is such that the Hecke algebra ${\cal H}_{n}(q)$ is semisimple,* then*,* as an ${\cal H}_{n}(q)$-module*,**

[TABLE]

Proof. In this case the three maps $c,d,e$ of Lemma 4.1 are bijective. Hence so too is the map $\mathop{\rm ch}\nolimits:{\rm Rep}^{1}\mskip 1.0mu{\cal H}_{n}(q)\to\mathop{\rm Sym}\nolimits_{n}$. The two modules in the statement are isomorphic since they have the same image in $\mathop{\rm Sym}\nolimits_{n}$.   $\mathchar 9219\relax$

5. The Hilbert series of the intertwining algebras

Let $K$ be any commutative ring. Denote by $U_{\mskip-2.0muK}$ the multiplicative subgroup of the ring $K[[t]]$ consisting of all formal power series with constant term equal to 1. In other words, $U_{\mskip-2.0muK}=1+{{m}}$ where ${{m}}$ is the ideal of $K[[t]]$ generated by $t$. There is a well known $\lambda$-ring structure on $U_{\mskip-2.0muK}$ [2, Lemma 1.1]. Addition in this ring is given by the usual multiplication of power series, while the ring multiplication is another binary operation $\circ$ which has the property that $(1+at)\circ(1+bt)=1+abt$ for $a,b\in K$. However, Theorem 5.5 makes use of a different multiplication $\diamond$ which gives an isomorphic ring structure on $U_{\mskip-2.0muK}$ and satisfies

[TABLE]

Now we will give a formal definition of this binary operation using the theory of symmetric functions. For each $n\geq 0$ extend the scalar product on $\mathop{\rm Sym}\nolimits_{n}$ to a symmetric $K$-bilinear form on the $K$-module $\mathop{\rm Sym}\nolimits_{n,K}=K\otimes_{{{Z}}}\mathop{\rm Sym}\nolimits_{n}$. Since this bilinear form induces a bijection

[TABLE]

to each $f\in U_{\mskip-2.0muK}$ there corresponds a uniquely determined element $\xi_{n}(f)\in\mathop{\rm Sym}\nolimits_{n,K}$ such that

[TABLE]

where ${\hat{f}}:\mathop{\rm Sym}\nolimits\to K$ is the ring homomorphism defined in section 3 and we identify elements $u\in\mathop{\rm Sym}\nolimits_{n}$ with their images $1\otimes u$ in $\mathop{\rm Sym}\nolimits_{n,K}$. Put

[TABLE]

Denote by $c_{n}(F)$ the coefficient of $t^{n}$ in a formal power series $F$. Considering the extensions of ${\hat{f}}$, ${\hat{g}}$ to $K$-algebra homomorphisms $K\otimes_{{Z}}\mathop{\rm Sym}\nolimits\to K$, we have

[TABLE]

Lemma 5.1. The formula $\xi(f)=\sum_{n=0}^{\infty}\xi_{n}(f)t^{n}$ defines a group homomorphism $\xi:U_{\mskip-2.0muK}\to U_{\!K\otimes_{{Z}}{}\mathop{\rm Sym}\nolimits}$.

Proof. Since ${\hat{f}}(1)=1$, it follows that $\xi_{0}(f)=1$, i.e. the map $\xi$ has values in the group $U_{\!K\otimes_{{Z}}{}\mathop{\rm Sym}\nolimits}$. We have to show that $\xi(fg)=\xi(f)\xi(g)$ for all $f,g\in U_{\mskip-2.0muK}$. Recall the Hopf algebra structure on $\mathop{\rm Sym}\nolimits$ (see [21, Ch. I, Example 5.25]). Let

[TABLE]

be the comultiplication in $\mathop{\rm Sym}\nolimits$ and $m_{K}:K\otimes_{{Z}}K\to K$ the multiplication in $K$. The formula

[TABLE]

defines a ring homomorphism $\mathop{\rm Sym}\nolimits\to K$. Since $\Delta(h_{n})=\sum_{i=0}^{n}h_{i}\otimes h_{n-i}$, we have

[TABLE]

for each $n\geq 0$. It follows that $\psi={\hat{p}}$ where $p=fg$. Hence

[TABLE]

for each $u\in\mathop{\rm Sym}\nolimits_{n}$. The last equality in this formula follows from the fact that $\Delta$ is the adjoint of the multiplication in $\mathop{\rm Sym}\nolimits$ with respect to the scalar product. The middle equality is true because $\Delta(u)\in\sum_{j=0}^{n}\mathop{\rm Sym}\nolimits_{j}\otimes_{{Z}}\mathop{\rm Sym}\nolimits_{n-j}$ and

[TABLE]

for any pair of elements $u^{\prime}\in\mathop{\rm Sym}\nolimits_{j}$ and $u^{\prime\prime}\in\mathop{\rm Sym}\nolimits_{n-j}$.

We conclude that $\xi_{n}(p)=\sum_{i=0}^{n}\xi_{i}(f)\xi_{n-i}(g)$ for each $n\geq 0$, which amounts to the desired equality of formal power series $\xi(p)=\xi(f)\xi(g)$.   $\mathchar 9219\relax$

Lemma 5.2. Let $f,g_{1},g_{2}\in U_{\mskip-2.0muK}$. Then $f\diamond(g_{1}g_{2})=(f\diamond g_{1})(f\diamond g_{2})$.

Proof. Since $\xi_{n}(g_{1}g_{2})=\sum_{i=0}^{n}\xi_{i}(g_{1})\xi_{n-i}(g_{2})$ by Lemma 5.1, we have

[TABLE]

Thus the two formal power series in question have equal coefficients.   $\mathchar 9219\relax$

Lemma 5.3. Let $a\in K$. If $g=(1-at)^{-1},$ then $f\diamond g=f(at)$. If $g=1+at,$ then $f\diamond g=f(-at)^{-1}$.

Proof. It follows from Lemma 5.2 that $f\diamond g^{-1}=(f\diamond g)^{-1}$ for all $g\in U_{\mskip-2.0muK}$. So the second assertion of Lemma 5.3 is equivalent to the first one. Let $g=(1-at)^{-1}$. Then ${\hat{g}}(h_{n})=a^{n}$ for all $n\geq 0$. Hence ${\hat{g}}(h_{\nu})=a^{|\nu|}$ for all $\nu\in{\cal P}$. Since the two ${{Z}}$-bases $\{h_{\nu}\}$ and $\{m_{\nu}\}$ of $\mathop{\rm Sym}\nolimits$ are dual to each other, we get

[TABLE]

Therefore, c_{n}(f\diamond g)={\hat{f}}\bigl{(}\xi_{n}(g)\bigr{)}=a^{n}{\hat{f}}(h_{n})=a^{n}c_{n}(f), which is the coefficient of $t^{n}$ in $f(at)$.   $\mathchar 9219\relax$

**Remark. **If $p=f\diamond g$, then ${\hat{p}}=m_{K}\circ({\hat{f}}\otimes{\hat{g}})\circ\Delta^{*}$ where $\Delta^{*}:\mathop{\rm Sym}\nolimits\to\mathop{\rm Sym}\nolimits\otimes_{{Z}}\mathop{\rm Sym}\nolimits$ is the ring homomorphism discussed in [21, Ch. I, Example 7.20]. Associativity of $\diamond$ follows from coassociativity of $\Delta^{*}$.

Lemma 5.4. Let $f$ be the Hilbert series of the $R$-symmetric algebra ${{S}}(V,R)$. If $R$ satisfies the $1$-dimensional source condition, then $\xi_{n}(f)=\mathop{\rm ch}\nolimits(V^{\otimes n})$ where $\mathop{\rm ch}\nolimits$ is the map of Proposition 4.2.

Proof. Here $K={{Z}}$, and so $\xi_{n}(f)\in\mathop{\rm Sym}\nolimits_{n}$. As we have seen in the proof of Theorem 4.5,

[TABLE]

for all $\nu\in{\cal P}(n)$. Hence $\langle\mathop{\rm ch}\nolimits(V^{\otimes n}),u\rangle={\hat{f}}(u)$ for all $u\in\mathop{\rm Sym}\nolimits_{n}$, and the conclusion follows from the definition of $\xi_{n}(f)$.   $\mathchar 9219\relax$

Suppose now that $V^{\prime}$ is a second finite dimensional vector space over the same field $\mathchar 1404\relax$, and $R^{\prime}$ is a Hecke symmetry on $V^{\prime}$ with the same parameter $q$ as the Hecke symmetry $R$ on $V$. We will define the algebra $A(R^{\prime},R)$ (cf. [17] and [24]).

Consider the tensor algebra of the vector space ${\mathop{\rm Hom}\nolimits\nolimits_{\mskip 2.0mu\mathchar 1404\relax}(V,V^{\prime})}^{\mskip-1.0mu*}$ dual to $\mathop{\rm Hom}\nolimits\nolimits_{\mskip 2.0mu\mathchar 1404\relax}(V,V^{\prime})$. Its homogeneous component of degree $n$ admits the following realization via canonical $\mathchar 1404\relax$-linear bijections

[TABLE]

Denote by ${\cal R}$ the linear operator on the vector space \mathop{\rm Hom}\nolimits\nolimits_{\mskip 2.0mu\mathchar 1404\relax}\bigl{(}V^{\otimes 2},{V^{\prime}}^{\mskip 2.0mu\otimes 2}\bigr{)}^{\mskip-2.0mu*} such that the dual operator ${\cal R}^{*}$ on $\mathop{\rm Hom}\nolimits\nolimits_{\mskip 2.0mu\mathchar 1404\relax}(V^{\otimes 2},{V^{\prime}}^{\mskip 2.0mu\otimes 2})$ is defined by the formula

[TABLE]

In other words, ${\cal R}={({R^{\prime}}^{\mskip 2.0mu*})}^{\mskip-2.0mu-1}\otimes R$ under the canonical identification

[TABLE]

The algebra $A(R^{\prime},R)$ is defined as the factor algebra of {{T}}\bigl{(}{\mathop{\rm Hom}\nolimits\nolimits_{\mskip 2.0mu\mathchar 1404\relax}(V,V^{\prime})}^{\mskip-1.0mu*}\bigr{)} by the homogeneous ideal $I$ generated by

[TABLE]

Under the canonical pairings between {{T}}_{n}\bigl{(}{\mathop{\rm Hom}\nolimits\nolimits_{\mskip 2.0mu\mathchar 1404\relax}(V,V^{\prime})}^{\mskip-2.0mu*}\bigr{)} and \mathop{\rm Hom}\nolimits\nolimits_{\mskip 2.0mu\mathchar 1404\relax}\bigl{(}V^{\otimes n},{V^{\prime}}^{\mskip 2.0mu\otimes n}\bigr{)} we have

[TABLE]

for $n=2$. In degree $n>2$ the homogeneous component of the ideal $I$ is

[TABLE]

Consider $V^{\otimes n}$ and ${V^{\prime}}^{\mskip 2.0mu\otimes n}$ as ${\cal H}_{n}(q)$-modules with respect to representations arising from the Hecke symmetries $R$ and $R^{\prime}$, respectively. It follows then that

[TABLE]

This equality holds for all $n\geq 0$ since $I_{n}=0$ when $n\leq 1$. Hence

[TABLE]

Note that $A(R^{\prime},R)=A(R)$ when $R^{\prime}=R$. Another graded algebra $E(R^{\prime},R)$ related to $A(R^{\prime},R)$ is defined as the factor algebra of {{T}}\bigl{(}{\mathop{\rm Hom}\nolimits\nolimits_{\mskip 2.0mu\mathchar 1404\relax}(V,V^{\prime})}^{\mskip-1.0mu*}\bigr{)} by the ideal generated by $\mathop{\rm Ker}\,({\cal R}-\mathop{\rm Id}\nolimits)$.

Theorem 5.5. Let $R,$ $R^{\prime}$ be Hecke symmetries on finite dimensional vector spaces $V,$ $V^{\prime}$ with the same parameter $q$ of the Hecke relation. Suppose that both $R$ and $R^{\prime}$ satisfy the $1$-dimensional source condition. Let

[TABLE]

be the Hilbert series of the algebras ${{S}}(V,R)$ and ${{S}}(V^{\prime}\!,R^{\prime})$. Then the Hilbert series of the algebra $A(R^{\prime},R)$ equals

[TABLE]

Proof. The isomorphism classes of ${\cal H}_{n}(q)$-modules $V^{\otimes n}$ and ${V^{\prime}}^{\mskip 2.0mu\otimes n}$ define elements of ${\rm Rep}^{1}\mskip 1.0mu{\cal H}_{n}(q)$. We have $\xi_{n}(f)=\mathop{\rm ch}\nolimits(V^{\otimes n})$ and $\xi_{n}(g)=\mathop{\rm ch}\nolimits({V^{\prime}}^{\mskip 2.0mu\otimes n})$ by Lemma 5.4. Now

[TABLE]

Proposition 4.2(v) and Lemma 5.4 yield

[TABLE]

Hence the Hilbert series \sum_{n\geq 0}\bigl{(}\dim A_{n}(R^{\prime},R)\bigr{)}t^{n} of the algebra $A(R^{\prime},R)$ coincides with $f\diamond g$. Taking $K={{R}}$ and applying Lemmas 5.2 and 5.3, we get

[TABLE]

which gives the required formula.   $\mathchar 9219\relax$

6. Even symmetries satisfying the trivial source condition

If $R$ is a Hecke symmetry of birank $(r_{0},r_{1})$ with $r_{1}=0$, then $R$ is said to have rank $r=r_{0}$. In the situation of Theorem 3.8 this happens precisely when the Hilbert series of $\Lambda(V,R)$ is a polynomial of degree $r$, i.e. the algebra $\Lambda(V,R)$ is finite dimensional with the grading of length $r$. In [15] Gurevich calls a closed Hecke symmetry of rank $r$ even. We do not need the assumption of closedness, however. Put

[TABLE]

Theorem 6.1. Suppose that $R$ is a Hecke symmetry of rank $r$ satisfying the trivial source condition. Let $\textstyle\prod\limits_{i=1}^{r}(1+\alpha_{i}t)$ be the Hilbert series of $\Lambda(V,R)$. For each $n\geq 0$

(i) V^{\otimes n}\cong\bigoplus_{\lambda\in{\cal P}(n)}\bigl{(}{\cal H}_{n}(q)\otimes_{{\cal H}_{\lambda}(q)}\mathchar 1404\relax_{\rm triv}\bigr{)}^{m_{\lambda}(\alpha)} as ${\cal H}_{n}(q)$-modules,**

*(ii) *the algebra $A_{n}(R)^{*}$ is Morita equivalent to the $q$-Schur algebra $S_{q}(r,n),$

(iii) $\{[V^{\lambda}]\mid\lambda\in{\cal P}(n,r)\}$ is a ${{Z}}$-basis of $\mathop{\rm Grot}\nolimits_{n}(R)$.

There is an isomorphism of graded rings $\mathop{\rm Grot}\nolimits(R)\cong\mathop{\rm Sym}\nolimits/I_{r}$ where $I_{r}$ is the ideal of $\mathop{\rm Sym}\nolimits$ with a ${{Z}}$-basis $\{s_{\lambda}\mid\lambda\in{\cal P},\ \ell(\lambda)>r\}$.

Proof. Under present assumptions the ${\cal H}_{n}(q)$-module $V^{\otimes n}$ defines an element of the group $\mathop{\rm Triv}{\cal H}_{n}(q)$, and so too does the ${\cal H}_{n}(q)$-module in the right hand side of (i). By Theorem 4.5

[TABLE]

Parts (iv) and (vi) of Proposition 4.2 show that the second module has the same image under the isomorphism of groups $\,\mathop{\rm ch}\nolimits:\mathop{\rm Triv}{\cal H}_{n}(q)\to\mathop{\rm Sym}\nolimits_{n}$, and therefore the two modules are isomorphic. This is conclusion (i).

Recall that $A_{n}(R)^{*}\cong E_{n}=\mathop{\rm End}\nolimits_{{\cal H}_{n}(q)}V^{\otimes n}$, while $S_{q}(r,n)$ is the endomorphism ring of the ${\cal H}_{n}(q)$-module

[TABLE]

where the sum is taken over all weak compositions of $n$ with $r$ components. A weak composition is allowed to have zero components $\mu_{i}$, and the corresponding parabolic subalgebra ${\cal H}_{\mu}(q)$ is the same as one would get by removing from $\mu$ all its zero components. If $\mu^{\prime\prime}\in{\cal P}(n,r)$ is the partition obtained from $\mu$ by rearranging its nonzero components in decreasing order, then

[TABLE]

On the other hand, $m_{\lambda}(\alpha)\geq 0$ for each $\lambda\in{\cal P}(n)$, and $m_{\lambda}(\alpha)=0$ if and only if $\ell(\lambda)>r$. Therefore nonzero summands in the right hand side of (i) correspond precisely to partitions $\lambda\in{\cal P}(n,r)$. We see that the ${\cal H}_{n}(q)$-modules $V^{\otimes n}$ and $M(r,n)$ have the same set of isomorphism classes of indecomposable direct summands. But then the endomorphism rings $E_{n}$ and $S_{q}(r,n)$ of the two modules are Morita equivalent. This entails (ii).

Although the direct sum decomposition of (i) in general does not come from an action of a torus, we can still use it to describe the structure of $A(R)$-comodules by means of a kind of weight spaces. Recall that right $A_{n}(R)$-comodules may be identified with left $E_{n}$-modules. Fix an isomorphism in (i). It gives a collection of ${\cal H}_{n}(q)$-submodules

[TABLE]

such that $V^{\otimes n}=\bigoplus M_{\lambda,\mskip 1.0mui}$ and $M_{\lambda,\mskip 1.0mui}\cong{\cal H}_{n}(q)\otimes_{{\cal H}_{\lambda}(q)}\mathchar 1404\relax_{\rm triv}$ for each pair $(\lambda,i)$. Denote by $\xi_{\lambda,\mskip 1.0mui}$ the projection onto $M_{\lambda,\mskip 1.0mui}$ with respect to this decomposition. Then

[TABLE]

is a set of pairwise orthogonal idempotents in the ring $E_{n}$ with $\sum\xi_{\lambda,\mskip 1.0mui}=1$. If $X$ is any left $E_{n}$-module, then $X=\bigoplus X_{\lambda,\mskip 1.0mui}$ where $X_{\lambda,\mskip 1.0mui}=\xi_{\lambda,\mskip 1.0mui}X$. Put $X_{\lambda}=X_{\lambda,1}$ and define the character of $X$ by the formula

[TABLE]

where $\mathop{\rm Sym}\nolimits_{n}(r)$ is the group of all symmetric homogeneous polynomials of degree $n$ in $r$ indeterminates $x_{1},\ldots,x_{r}$.

Since $M_{\lambda,\mskip 1.0mui}\cong M_{\lambda,1}$, the idempotent $\xi_{\lambda,\mskip 1.0mui}$ is conjugate to $\xi_{\lambda,1}$ by an inner automorphism of $E_{n}$. Hence $\dim X_{\lambda,\mskip 1.0mui}=\dim X_{\lambda}$ for all $i$ such that $1\leq i\leq m_{\lambda}(\alpha)$. In particular, it follows that $X=0$ whenever $\mathop{\rm ch}\nolimits^{\mskip-1.0muR}(X)=0$.

If $\,\,0\to X^{\prime}\to X\to X^{\prime\prime}\to 0\,\,$ is an exact sequence of finite dimensional left $E_{n}$-modules, then $\,\,\dim X_{\lambda}=\dim X^{\prime}_{\lambda}+\dim X^{\prime\prime}_{\lambda}\,\,$ for each $\lambda$, whence

[TABLE]

So $\mathop{\rm ch}\nolimits^{\mskip-1.0muR}$ gives rise to a group homomorphism

[TABLE]

which will be denoted by the same symbol $\mathop{\rm ch}\nolimits^{\mskip-1.0muR}$.

It is well known that the Schur polynomials $s_{\lambda}(x_{1},\ldots,x_{r})$ with $\lambda\in{\cal P}(n,r)$ form a ${{Z}}$-basis of $\mathop{\rm Sym}\nolimits_{n}(r)$. On the other hand, isomorphism classes of simple $S_{q}(n,r)$-modules are also parametrized by the set ${\cal P}(n,r)$. The Morita equivalent algebra $E_{n}$ has the same number of simple modules equal to the cardinality of ${\cal P}(n,r)$. In other words, $\mathop{\rm Grot}\nolimits_{n}(R)$ and $\mathop{\rm Sym}\nolimits_{n}(r)$ are free abelian groups of equal ranks. As will be proved separately in Lemma 6.2, $s_{\lambda}(x_{1},\ldots,x_{r})$ is the image of $V^{\lambda}$ under $\mathop{\rm ch}\nolimits^{\mskip-1.0muR}\mskip-2.0mu$. Hence $\mathop{\rm ch}\nolimits^{\mskip-1.0muR}$ is surjective, but then $\mathop{\rm ch}\nolimits^{\mskip-1.0muR}$ has to map the group $\mathop{\rm Grot}\nolimits_{n}(R)$ isomorphically onto $\mathop{\rm Sym}\nolimits_{n}(r)$, and (iii) is also clear.

The ring homomorphism $\varphi:\mathop{\rm Sym}\nolimits\to\mathop{\rm Grot}\nolimits(R)$ defined in Proposition 3.2 sends $s_{\lambda}$ to $[V^{\lambda}]$ for each $\lambda\in{\cal P}$. It follows from (iii) that $\varphi$ maps the subgroup of $\mathop{\rm Sym}\nolimits$ generated by $\{s_{\lambda}\mid\ell(\lambda)\leq r\}$ isomorphically onto $\mathop{\rm Grot}\nolimits(R)$. In particular, $\varphi$ is surjective. If $\ell(\lambda)>r$, then $V^{\lambda}=0$ by Corollary 3.10. This can be also seen from the fact that $\,\mathop{\rm ch}\nolimits^{\mskip-1.0muR}(V^{\lambda})=0\,$: assuming that $|\lambda|=n$, we have

[TABLE]

Recall that $K_{\lambda\mu}=0$ unless $\mu\leq\lambda$ with respect to the dominance order. If $\mu\leq\lambda$, then $\ell(\mu)\geq\ell(\lambda)>r$, in which case $m_{\mu}(x_{1},\ldots,x_{r})=0$. Thus all summands in the above expression vanish.

It follows that $\,\mathop{\rm Ker}\varphi=I_{r}\,$, and we are done.   $\mathchar 9219\relax$

Lemma 6.2. For each $\lambda\in{\cal P}(n)$ we have $\mathop{\rm ch}\nolimits^{\mskip-1.0muR}(V^{\lambda})=s_{\lambda}(x_{1},\ldots,x_{r})$.

Proof. First we evaluate $\mathop{\rm ch}\nolimits^{\mskip-1.0muR}(\mskip 2.0mu{{S}}{}^{\mskip 0.5mu\mu})$ for $\mu\in{\cal P}(n)$. Recall that $\,{{S}}{}^{\mskip 0.5mu\mu}\cong\mathchar 1404\relax_{\rm triv}\otimes_{{\cal H}_{\mu}(q)}V^{\otimes n}$. If $\lambda\in{\cal P}(n,r)$, then

[TABLE]

since the $E_{n}$-module structure on ${{S}}{}^{\mskip 0.5mu\mu}$ comes from the action of $E_{n}$ on $V^{\otimes n}$. Hence $\dim{{S}}{}^{\mskip 0.5mu\mu}_{\mskip 1.0mu\lambda}$ is equal to the number $N_{\mu\lambda}$ of all ${{S}}_{\mu}\,$-$\,{{S}}_{\lambda}$ double cosets in the group ${{S}}_{n}$. There is also a combinatorial description of $N_{\mu\lambda}=N_{\lambda\mu}$ as the number of all nonnegative integer matrices of size $\ell(\lambda)\times\ell(\mu)$ having row sums $\lambda_{i}$, $\,i=1,\ldots,\ell(\lambda)$, and column sums $\mu_{j}$, $\,j=1,\ldots,\ell(\mu)$. By [21, Ch. I, (6.7)] $\,h_{\mu}=\sum_{\lambda\in{\cal P}(n)}N_{\mu\lambda}\,m_{\lambda}$. If $\ell(\lambda)>r$, then $m_{\lambda}(x_{1},\ldots,x_{r})=0$. Therefore

[TABLE]

Let $\varphi:\mathop{\rm Sym}\nolimits\to\mathop{\rm Grot}\nolimits(R)$ be the ring homomorphism defined in Proposition 3.2. Since $\varphi(h_{\mu})=[\,{{S}}{}^{\mskip 0.5mu\mu}]$, the previous equality can be rewritten as

[TABLE]

for $u=h_{\mu}$. Since the set $\{h_{\mu}\mid\mu\in{\cal P}(n)\}$ generates the whole group $\mathop{\rm Sym}\nolimits_{n}$, the formula above holds then for all $u\in\mathop{\rm Sym}\nolimits_{n}$. Taking $u=s_{\lambda}$, we get the required conclusion.   $\mathchar 9219\relax$

**Remark. **For each $\lambda\in{\cal P}(n,r)$ it follows from Lemma 6.2 that $\dim\,(V^{\lambda})_{\lambda}=1$ and $(V^{\lambda})_{\mu}=0$ unless $\mu\leq\lambda$. In this sense $\lambda$ is the highest weight of $V^{\lambda}$. Since the function $X\mapsto\dim X_{\lambda}$ is additive on exact sequences of $E_{n}$-modules, there is exactly one composition factor $L^{\lambda}$ of $V^{\lambda}$ with nonzero $\lambda$-weight space. Clearly $L^{\lambda}$ is also a module of highest weight $\lambda$. In particular, $L^{\lambda}\not\cong L^{\mu}_{\vphantom{\lambda}}$ whenever $\lambda,\mu\in{\cal P}(n,r)$ and $\lambda\neq\mu$. This implies that $\{L^{\lambda}\mid\lambda\in{\cal P}(n,r)\}$ is the full set of pairwise nonisomorphic simple $E_{n}$-modules.

Let $H(R)$ stand for the Hopf envelope of $A(R)$. If $R$ and $R^{\prime}$ are two closed Hecke symmetries with the same parameter $q$ and of the same birank $(r_{0},r_{1})$, then in the semisimple case, by a theorem of Phùng Hô Hai [18, Th. 4.3], there is a braided monoidal equivalence between the categories of right $H(R)$-comodules and right $H(R^{\prime})$-comodules.

We are interested in the bialgebra version of this result. Let $R$ and $R^{\prime}$ be not necessarily closed Hecke symmetries on vector spaces $V$ and $V^{\prime}$, respectively, with the same parameter $q$ of the Hecke relation. Consider the ${\cal H}_{n}(q)$-module structures on $V^{\otimes n}$ and ${V^{\prime}}^{\mskip 2.0mu\otimes n}$ arising from $R$ and $R^{\prime}$. It was proved in [24, Th. 7.2] that there is a braided monoidal equivalence between the categories of right $A(R)$-comodules and right $A(R^{\prime})$-comodules provided that for each $n>0$ the indecomposable ${\cal H}_{n}(q)$-modules isomorphic to direct summands of $V^{\otimes n}$ are the same as those isomorphic to direct summands of ${V^{\prime}}^{\mskip 2.0mu\otimes n}$.

If $R$ and $R^{\prime}$ have the same birank $(r_{0},r_{1})$, then in the semisimple case the required condition on direct summands is satisfied since in each of the two ${\cal H}_{n}(q)$-modules the simple submodules are precisely the Specht modules $S^{\lambda}$ with $\lambda\in\Gamma(r_{0},r_{1})\cap{\cal P}(n)$. Unfortunately we are not able to extend this result to the nonsemisimple case because Theorem 4.5 does not provide enough information on the module structure of the tensor powers. Therefore we have to restrict the class of Hecke symmetries in the following statement:

Theorem 6.3. Let $R$ and $R^{\prime}$ be Hecke symmetries with the same parameter $q,$ of the same rank $r$,* and both satisfying the trivial source condition. Then there is a braided monoidal equivalence between the categories of right (or left) $A(R)$-comodules and $A(R^{\prime})$-comodules.*

Proof. By Theorem 6.1 the ${\cal H}_{n}(q)$-modules $V^{\otimes n}$ and ${V^{\prime}}^{\mskip 2.0mu\otimes n}$ have the same set of isomorphism classes of indecomposable direct summands. So [24, Th. 7.2] does apply.   $\mathchar 9219\relax$

In conclusion we improve yet another result from [24]. Definitions of the algebras $A(R^{\prime},R)$ and $E(R^{\prime},R)$ have been recalled in section 5.

Theorem 6.4. Let $R$ and $R^{\prime}$ be Hecke symmetries with the same parameter $q,$ both satisfying the trivial source condition. Suppose that the algebras $\Lambda(V,R)$ and $\Lambda(V^{\prime}\!,R^{\prime})$ are Frobenius with the gradings of length $r$ and $r^{\prime},$ respectively. Then the algebra $E(R^{\prime},R)$ is Frobenius with the grading of length $rr^{\prime},$ while $A(R^{\prime},R)$ is Gorenstein of global dimension $rr^{\prime}$.

Proof. The Hilbert series of the algebras $\Lambda(V,R)$ and $\Lambda(V^{\prime}\!,R^{\prime})$ are polynomials of degree $r$ and $r^{\prime}$, respectively. We can write them as

[TABLE]

The Hilbert series of ${{S}}(V,R)$ and ${{S}}(V^{\prime}\!,R^{\prime})$ are $\prod\mskip 1.0mu(1-\alpha_{i}t)^{-1}$ and $\prod\mskip 1.0mu(1-\alpha^{\prime}_{j}t)^{-1}$ by Theorem 3.8. For the graded algebras $A(R^{\prime},R)$ and $E(R^{\prime},R)$ it holds then

[TABLE]

The first formula here follows from Theorem 5.5. The second formula is a consequence of the relation $\,H_{E(R^{\prime}\!,R)}(t)\,H_{A(R^{\prime}\!,R)}(-t)=1\,$ proved in [24, Th. 6.2].

Since $\dim\Lambda_{r}(V,R)=\dim\Lambda_{r^{\prime}}(V^{\prime}\!,R^{\prime})=1$, we have $\prod\alpha_{i}=\prod\alpha^{\prime}_{j}=1$. Thus $H_{E(R^{\prime}\!,R)}$ is a polynomial of degree $rr^{\prime}$ with the leading coefficient equal to 1. The conclusion follows then from [24, Th. 6.6].   $\mathchar 9219\relax$

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