The Koszul map K

TL;DR

This paper introduces an equivariant isomorphism linking the universal enveloping algebra of gl(n) with polynomial functions on matrices, generalizing Koszul's classical results and sharpening the PBW theorem.

Contribution

It constructs a linear equivariant isomorphism between U(gl(n)) and polynomial matrix algebra, extending Koszul's Capelli determinant results and refining the PBW isomorphism.

Findings

Maps Capelli bitableaux to determinantal bitableaux

Establishes an isomorphism that generalizes Koszul's theorem

Provides a sharpened PBW isomorphism for U(gl(n))

Abstract

The Bitableax correspondence isomorphism/Koszul map Theorem (BCK Theorem, for short, Theorem 6.5 below) describes a relevant pair of mutually inverse vector space isomorphisms, the Koszul map K : U(gl(n))-> Sym(gl(n)) and the bitableaux correspondence iWe describe a linear \emph{equivariant isomorphism} $\mathcal{K}$ from the enveloping algebra $\mathbf{U}(gl(n))$ to the algebra ${\mathbb C}[M_{n,n}] \cong \mathbf{Sym}(gl(n))$ of polynomials in the entries of a ``generic'' square matrix of order $n$. The isomorphism $\mathcal{K}$ maps any {\textit{Capelli bitableau}} $[S|T]$ in $\mathbf{U}(gl(n))$ to the {\textit{(determinantal) bitableau}} $(S|T)$ in ${\mathbb C}[M_{n,n}]$ and any {\textit{Capelli *-bitableau}} $[S|T]^*$ in $\mathbf{U}(gl(n))$ to the {\textit{(permanental) *-bitableau}} $(S|T)^*$ in ${\mathbb C}[M_{n,n}]$. These results are far-reaching generalizations of the pioneering result of J.-L. Koszul [19] on the Capelli determinant in $\mathbf{U}(gl(n))$ (see, e.g. [24], [27]). We introduce {\textit{column}} Capelli bitableaux and *-bitableaux in Section 6; since they are mapped by the isomorphism $\mathcal{K}$ to {\textit{monomials}} in ${\mathbb C}[M_{n,n}]$, this isomorphism can be regarded as a sharpened version of the PBW isomorphism for the enveloping algebra $\mathbf{U}(gl(n))$. Since the center $\boldsymbol{\zeta}(n)$ of $\mathbf{U}(gl(n))$ equals the subalgebra of invariants $\mathbf{U}(gl(n))^{Ad_{gl(n)}}$, then $$ \mathcal{K} \big[ \boldsymbol{\zeta}(n) \big] = {\mathbb C}[M_{n,n}]^{ad_{gl(n)}}. $$somorphism B : Sym(gl(n)) -> U(gl(n)) that deeply link the enveloping algebra U(gl(n)) of the general linear Lie algebra gl(n) and the symmetric algebra Sym(gl(n)). The BCK Theorem can be regarded as a sharpened version of the PBW Theorem for the enveloping algebra U(gl(n)).

Equations

$$\mathcal{K}\big{[}\boldsymbol{\zeta}(n)\big{]}={\mathbb{C}}[M_{n,n}]^{ad_{gl(n)}}.$$

$$\mathcal{K}:\mathbf{U}(gl(n))\to \mathbb{C}[M_{n,n}]\cong \mathbf{Sym}(gl(n))$$

$$\mathcal{K}\big{[}\boldsymbol{\zeta}(n)\big{]}={\mathbb{C}}[M_{n,n}]^{ad_{gl(n)}},$$

$$\mathbb{C}[M_{n,n}]=\mathbb{C}[(i|j){]}_{i,j=1,\dots ,n}$$

$$M_{n,n}=\left[(i|j)\right]_{i,j=1,\ldots,n}=\left(\begin{array}[]{ccc}(1|1)&\ldots&(1|n)\\ \vdots&&\vdots\\ (n|1)&\ldots&(n|n)\\ \end{array}\right).$$

$$(\omega |\varpi )=(i_1{i}_2\cdots i_p|j_1{j}_2\cdots j_p)$$

$$(\omega|\varpi)=(-1)^{p\choose 2}\ det\Big{(}\ (i_{r}|j_{s})\ \Big{)}_{r,s=1,2,\ldots,p}\in{\mathbb{C}}[M_{n,n}].$$

$$(S|T)=\left(\begin{array}[]{c}\omega_{1}\\ \omega_{2}\\ \vdots\\ \omega_{p}\end{array}\right|\left.\begin{array}[]{c}\varpi_{1}\\ \varpi_{2}\\ \vdots\\ \varpi_{p}\end{array}\right)$$

$$(S|T)=\text{s}({\omega }_1|{\varpi }_1)({\omega }_2|{\varpi }_2)\cdots ({\omega }_p|{\varpi }_p),$$

$$\text{s}=(-1{)}^{\ell ({\omega }_2)\ell ({\varpi }_1)+\ell ({\omega }_3)(\ell ({\varpi }_1)+\ell ({\varpi }_2))+\cdots +\ell ({\omega }_p)(\ell ({\varpi }_1)+\ell ({\varpi }_2)+\cdots +\ell ({\varpi }_{p-1}))},$$

$$(\omega |\varpi {)}^{\ast }=(i_1{i}_2\cdots i_p|j_1{j}_2\cdots j_p{)}^{\ast }$$

$$(\omega|\varpi)^{*}=\ per\Big{(}\ (i_{r}|j_{s})\ \Big{)}_{r,s=1,2,\ldots,p}\in{\mathbb{C}}[M_{n,n}].$$

$$(S|T)^{*}=\left(\begin{array}[]{c}\omega_{1}\\ \omega_{2}\\ \vdots\\ \omega_{p}\end{array}\right|\left.\begin{array}[]{c}\varpi_{1}\\ \varpi_{2}\\ \vdots\\ \varpi_{p}\end{array}\right)^{*}$$

$$(S|T{)}^{\ast }=({\omega }_1|{\varpi }_1{)}^{\ast }({\omega }_2|{\varpi }_2{)}^{\ast }\cdots ({\omega }_p|{\varpi }_p{)}^{\ast }.$$

$$\left(\begin{array}[]{c}i_{1}\\ i_{2}\\ \vdots\\ i_{h}\end{array}\right|\left.\begin{array}[]{c}j_{1}\\ j_{2}\\ \vdots\\ j_{h}\end{array}\right)=(-1)^{h\choose 2}(i_{1}|j_{1})(i_{2}|j_{2})\cdots(i_{h}|j_{h})$$

$$\left(\begin{array}[]{c}i_{1}\\ i_{2}\\ \vdots\\ i_{h}\end{array}\right|\left.\begin{array}[]{c}j_{1}\\ j_{2}\\ \vdots\\ j_{h}\end{array}\right)^{*}=(i_{1}|j_{1})(i_{2}|j_{2})\cdots(i_{h}|j_{h}).$$

$$(S|\framebox{$T$}))=\sum_{\overline{T}}\ (S|\overline{T}),$$

$$\left(\begin{array}[]{cc}1&3\\ 2&4\end{array}\right|\left.\framebox{$\begin{array}[]{cc}1&2\\ 1&3\end{array}$}\ \right)=2\left(\begin{array}[]{cc}1&3\\ 2&4\end{array}\right|\left.\begin{array}[]{cc}1&2\\ 1&3\end{array}\right)+2\left(\begin{array}[]{cc}1&3\\ 2&4\end{array}\right|\left.\begin{array}[]{cc}1&3\\ 1&2\end{array}\right).$$

$$gl(n)\rightarrow End_{{\mathbb{C}}}\big{[}{\mathbb{C}}[M_{n,n}]\big{]}$$

$${\rho }^{\text{l}}:e_{ij}\mapsto D_{ij}^{\text{l}}.$$

$${\rho }^{\text{r}}:e_{ij}\mapsto -D_{ji}^{\text{r}}.$$

$$a{d}_{gl(n)}:e_{ij}\mapsto D_{ij}^{\text{l}}-D_{ji}^{\text{r}}.$$

$$T_{ij}\big{(}\textbf{M}\big{)}=e_{ij}\textbf{M}-\textbf{M}e_{ij},$$

$$T_{ij}\big{(}e_{st}\big{)}=\delta_{js}e_{it}-\delta_{it}e_{sj}=\big{[}e_{ij},e_{st}\big{]}$$

$$T_{ij}\circ T_{hk}-T_{hk}\circ T_{ij}={\delta }_{jh}{T}_{ik}-{\delta }_{ik}{T}_{hj}.$$

$$Ad_{gl(n)}:gl(n)\rightarrow End_{{\mathbb{C}}}\big{[}\mathbf{U}(gl(n))\big{]}$$

$$e_{ij}\mapsto T_{ij}$$

$$W_0=V_{m_0},W_1=V_{m_1}\oplus V_n.$$

$$\{e_{a,b};a,b\in A_0\cup A_1\cup L\},|e_{a,b}|=|a|+|b|\in {\mathbb{Z}}_2$$

$$\left[e_{a,b},e_{c,d}\right]={\delta }_{bc}{e}_{a,d}-(-1{)}^{(|a|+|b|)(|c|+|d|)}{\delta }_{ad}{e}_{c,b},$$

Full text

The isomorphism $\mathcal{K}$ maps any Capelli bitableau $[S|T]$ in $\mathbf{U}(gl(n))$ to the (determinantal) bitableau $(S|T)$ in ${\mathbb{C}}[M_{n,n}]$ and any *Capelli -bitableau $[S|T]^{*}$ in $\mathbf{U}(gl(n))$ to the *(permanental) -bitableau $(S|T)^{*}$ in ${\mathbb{C}}[M_{n,n}]$.

These results are far-reaching generalizations of the pioneering result of J.-L. Koszul [19] on the Capelli determinant in $\mathbf{U}(gl(n))$ (see, e.g. [24], [27]).

We introduce column Capelli bitableaux and *-bitableaux in Section 6; since they are mapped by the isomorphism $\mathcal{K}$ to monomials in ${\mathbb{C}}[M_{n,n}]$, this isomorphism can be regarded as a sharpened version of the PBW isomorphism for the enveloping algebra $\mathbf{U}(gl(n))$.

Since the center $\boldsymbol{\zeta}(n)$ of $\mathbf{U}(gl(n))$ equals the subalgebra of invariants $\mathbf{U}(gl(n))^{Ad_{gl(n)}}$, then

[TABLE]

Keyword: Enveloping algebras; Young tableaux; Lie superalgebras;

central elements; Capelli determinants.

1 Introduction

The starting points of the present work are ([5], [6]):

• –

The linear operator $\mathcal{B}:{\mathbb{C}}[M_{n,n}]\rightarrow\mathbf{U}(gl(n))$ that maps any (determinantal) bitableau $(S|T)$ in ${\mathbb{C}}[M_{n,n}]$ to the Capelli bitableau $[S|T]$ in $\mathbf{U}(gl(n))$.

• –

The linear operator $\mathcal{B^{*}}:{\mathbb{C}}[M_{n,n}]\rightarrow\mathbf{U}(gl(n))$ that maps any (permanental) bitableau $(S|T)^{*}$ in ${\mathbb{C}}[M_{n,n}]$ to the *Capelli -bitableau $[S|T]^{*}$ in $\mathbf{U}(gl(n))$.

The map

[TABLE]

introduced by Koszul in 1981 [19] is proved to be the inverse of both $\mathcal{B}$ and $\mathcal{B}^{*}$. Then $\mathcal{B}$, $\mathcal{B}^{*}$, $\mathcal{K}$ are vector space isomorphisms and $\mathcal{B}=\mathcal{B}^{*}$.

Since the set of standard bitableaux is a basis of ${\mathbb{C}}[M_{n,n}]$ ([16], [15], [14], [17]), then the set of standard Capelli bitableaux is a basis of $\mathbf{U}(gl(n))$. Since the set of costandard *-bitableaux is a basis of ${\mathbb{C}}[M_{n,n}]$, then the set of costandard Capelli *-bitableaux is a basis of $\mathbf{U}(gl(n))$.

Some of these topics were treated in a sketchy way in the present author’s notes [5], [6] (in the more general setting of superalgebras), in a rather cumbersome notation and almost without proofs. The main novelty of the present approach is the major role played by column Capelli bitableaux and *column Capelli -bitableaux; although they are far from being “monomials” in the enveloping algebra $\mathbf{U}(gl(n))$, their images with respect to the Koszul isomorphism $\mathcal{K}$ are indeed monomials in the polynomial algebra ${\mathbb{C}}[M_{n,n}]$. Therefore, column Capelli bitableaux and column Capelli *-bitableaux play the same role in $\mathbf{U}(gl(n))$ that monomials play in ${\mathbb{C}}[M_{n,n}]$ and this leads to a new and transparent presentation.

The expressions of column Capelli bitableaux and column Capelli *-bitableaux as elements of $\mathbf{U}(gl(n))$ can be simply computed.

Capelli bitableaux and Capelli *-bitableaux expand - up to a global sign - into column Capelli bitableaux just in the same way as determinantal bitableaux and permanental *-bitableaux expand into the corresponding monomials in ${\mathbb{C}}[M_{n,n}]$ (Laplace expansions).

The isomorphism $\mathcal{B}=\mathcal{K}^{-1}$ maps any right symmetrized bitableau (S|\framebox{T})\in{\mathbb{C}}[M_{n,n}] ([3], [4]) to the right Young-Capelli bitableau [S|\framebox{T}] in $\mathbf{U}(gl(n))$. The basis of standard right Young-Capelli bitableaux acts in a remarkable way on the Gordan-Capelli basis of standard right symmetrized bitableaux. Moreover, the elements of the Schur-Sahi-Okounkov basis of the center $\boldsymbol{\zeta}(n)$ of $\mathbf{U}(gl(n))$ (quantum immanants [25], [21], [22], [23]) admit quite effective presentations as linear combinations of right Young-Capelli bitableaux as well as of Capelli immanants [8] and [7].

The Koszul map $\mathcal{K}$ is proved to be an equivariant isomorphism with respect to the adjoint representations of $gl(n)$ on $\mathbf{U}(gl(n))$ and ${\mathbb{C}}[M_{n,n}]$ (polarization action), respectively. Since the center $\boldsymbol{\zeta}(n)$ of $\mathbf{U}(gl(n))$ is the subalgebra of $Ad_{gl(n)}$-invariants of $\mathbf{U}(gl(n))^{Ad_{gl(n)}}$, then

[TABLE]

where ${\mathbb{C}}[M_{n,n}]^{ad_{gl(n)}}$ is the subalgebra of $ad_{gl(n)}$-invariants of ${\mathbb{C}}[M_{n,n}]$.

2 Determinantal Young bitableaux, permanental *Young -bitableaux

and right symmetrized bitableaux in the polynomial algebra ${\mathbb{C}}[M_{n,n}]$

Let

[TABLE]

be the polynomial algebra in the (commutative) “generic" entries $(i|j)$ of the matrix:

[TABLE]

Given the standard basis \big{\{}e_{ij};\ i,j=1,2,\ldots,n\big{\}} of the general linear Lie algebra $gl(n)$, the map $e_{ij}\mapsto(i|j)$ induces an isomorphism $\mathbf{Sym}(gl(n))\cong{\mathbb{C}}[M_{n,n}]$.

Let $\omega=i_{1}i_{2}\cdots i_{p}$, $\varpi=j_{1}j_{1}\cdots j_{p}$ be words on the alphabet $\{1,2,\ldots,n\}$.

Following [17] and [3], the biproduct of the two words $\omega$ and $\varpi$

[TABLE]

is the signed minor:

[TABLE]

Let $S=(\omega_{1},\omega_{2},\ldots,\omega_{p})$ and $T=(\varpi_{1},\varpi_{2},\ldots,\varpi_{p})$ be Young tableaux on $\{1,2,\ldots,n\}$ of the same shape $\lambda$.

Following again [17] and [3], the (determinantal) Young bitableau

[TABLE]

is the signed product of the biproducts of the pairs of corresponding rows:

[TABLE]

where

[TABLE]

and the symbol $\ell(w)$ denotes the length of the word $w$.

The *-biproduct of the two words $\omega$ and $\varpi$

[TABLE]

is the permanent:

[TABLE]

Let $S=(\omega_{1},\omega_{2},\ldots,\omega_{p})$ and $T=(\varpi_{1},\varpi_{2},\ldots,\varpi_{p})$ be Young tableaux on $\{1,2,\ldots,n\}$ of the same shape $\lambda$.

Following again [17] and [3], the (permanental) *Young -bitableau

[TABLE]

is the product of the *-biproducts of the pairs of corresponding rows:

[TABLE]

A column Young tableau of depth $h$ is a tableau of shape $(1^{h})$. Then for a column Young bitableau, we have:

[TABLE]

and for a column Young *-bitableau, we have:

[TABLE]

We recall the definition of the right symmetrized bitableau (S|\framebox{T})) (see, e.g. [3]):

[TABLE]

where the sum is extended over all $\overline{T}$ column permuted of $T$ (hence, repeated entries in a column give rise to multiplicities).

Example 2.1**.**

[TABLE]

We recall same elementary definitions and notational conventions. Given a partition (shape) $\lambda=(\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{p})\vdash n$, let $\widetilde{\lambda}=(\widetilde{\lambda}_{1},\widetilde{\lambda}_{2}\geq\cdots\geq\widetilde{\lambda}_{q})\vdash n$ denote its conjugate partition, where $\widetilde{\lambda}_{s}=\#\{t;\lambda_{t}\geq s\}$. Similarly, given a Young tableau $S$ of shape $sh(S)=\lambda$, let $\widetilde{S}$ denote its conjugate (dual) Young tableau. In plain words, $\widetilde{S}$ is the tableau whose rows are the columns of $S$ and whose shape is $sh(\widetilde{S})=\widetilde{\lambda}$. A Young tableau $X$ on the (linearly ordered) set $L=\{1<2<\cdots<n\}$ is said to be standard whenever its rows are increasing from left to right and its columns are non decreasing downwards. In a dual way, a Young tableau $Y$ is said to be costandard whenever its conjugate Young tableau $\widetilde{Y}$ is standard.

We recall the basis theorems for standard determinantal bitableaux (see, e.g. [16], [15], [14]), *costandard permanental -bitableaux [17] and right symmetrized bitableaux [3], respectively.

Proposition 2.2**.**

The sets

• –

\Big{\{}(S|T);\ sh(S)=sh(T)=\lambda,\ \lambda_{1}\leq n,\ S,T\ standard\Big{\}},

• –

\Big{\{}(U|V)^{*};\ sh(U)=sh(V)=\mu,\ \widetilde{\mu_{1}}\leq n,\ U,V\ costandard\Big{\}},

• –

\Big{\{}(S|\framebox{T});\ sh(S)=sh(T)=\lambda,\ \lambda_{1}\leq n,\ S,T\ standard\Big{\}}**

are linear bases of ${\mathbb{C}}[M_{n,n}]$.

3 Polarization operators and Lie algebra representations of $gl(n)$

on ${\mathbb{C}}[M_{n,n}]$ and $\mathbf{U}(gl(n))$

Given $i,j=1,2,\ldots,n$, the left polarization operator (of $j$ to $i$) $D^{\textit{l}}_{ij}$ is the linear operator from ${\mathbb{C}}[M_{n,n}]$ to itself defined by the conditions:

• –

$D^{\textit{l}}_{ij}$ is a derivation

• –

D^{\textit{l}}_{ij}\big{(}(h|k)\big{)}=\delta_{jh}(i|k) for every $k.$

Similarly, the right polarization operator (of $i$ to $j$) $D^{\textit{r}}_{ji}$ is the linear operator from ${\mathbb{C}}[M_{n,n}]$ to itself defined by the conditions:

• –

$D^{\textit{r}}_{ji}$ is a derivation

• –

D^{\textit{r}}_{ji}\big{(}(h|k)\big{)}=\delta_{ik}(h|j) for every $h$.

In the following, we consider three Lie algebra representations

[TABLE]

and the corresponding Lie modules.

The left (covariant) representation $\rho^{\textit{l}}$ is defined by setting

[TABLE] 2. 2.

The right (contravariant) representation $\rho^{\textit{r}}$ is defined by setting

[TABLE] 3. 3.

Notice that $D^{\textit{l}}_{ij}D^{\textit{r}}_{hk}=D^{\textit{r}}_{hk}D^{\textit{l}}_{ij}$. The adjoint representation $ad_{gl(n)}$ is defined by setting

[TABLE]

Given $i,j=1,2,\ldots,n$, consider the linear operator $T_{ij}$ from $\mathbf{U}(gl(n))$ to itself defined by setting

[TABLE]

for every $\textbf{M}\in\mathbf{U}(gl(n))$.

We recall that $T_{ij}$ is the unique derivation of $\mathbf{U}(gl(n))$ such that

[TABLE]

for every $s,t=1,2,\ldots,n$. Hence

[TABLE]

The Lie algebra representation

[TABLE]

[TABLE]

is the adjoint representation of $\mathbf{U}(gl(n))$ on itself.

4 The superalgebraic approach to the enveloping algebra $\mathbf{U}(gl(n))$

In this Section, we provide a synthetic presentation of the superalgebraic method of virtual variables for $gl(n)$.

This method was developed by the present authors for the general linear Lie superalgebras $gl(m|n)$ [18], in the series of notes [1], [2], [3], [4], [5], [6].

The technique of virtual variables is an extension of Capelli’s method of * variabili ausilarie* (Capelli [12], see also Weyl [27]).

Capelli introduced the technique of * variabili ausilarie* in order to manage symmetrizer operators in terms of polarization operators and to simplify the study of some skew-symmetrizer operators (namely, the famous central Capelli operator).

Capelli’s idea was well suited to treat symmetrization, but it did not work in the same efficient way while dealing with skew-symmetrization.

One had to wait the introduction of the notion of superalgebras (see,e.g. [26], [18]) to have the right conceptual framework to treat symmetry and skew-symmetry in one and the same way. To the best of our knowledge, the first mathematician who intuited the connection between Capelli’s idea and superalgebras was Koszul in $1981$ [19]. In particular, Koszul proved that the classical determinantal Capelli operator can be rewritten - in a much simpler way - by adding to the symbols to be dealt with an extra auxiliary symbol that obeys to different commutation relations.

The superalgebraic method of virtual variables allows us to express remarkable classes of elements in $\mathbf{U}(gl(n))$ as images - with respect to the Capelli devirtualization epimorphism - of simple monomials and to obtain transparent combinatorial descriptions of their actions on irreducible $gl(n)-$modules.

This method is very well suited for the study of the polarization action of $\mathbf{U}(gl(n))$ on ${\mathbb{C}}[M_{n,n}]$ and for the study of the center of $\mathbf{U}(gl(n))$.

4.1 The superalgebras ${\mathbb{C}}[M_{m_{0}|m_{1}+n,n}]$ and $gl(m_{0}|m_{1}+n)$

4.1.1 The general linear Lie super algebra $gl(m_{0}|m_{1}+n)$

Given a vector space $V_{n}$ of dimension $n$, we will regard it as a subspace of a $\mathbb{Z}_{2}-$graded vector space $W=W_{0}\oplus W_{1}$, where

[TABLE]

The vector spaces $V_{m_{0}}$ and $V_{m_{1}}$ (we assume that $dim(V_{m_{0}})=m_{0}$ and $dim(V_{m_{1}})=m_{1}$ are “sufficiently large”) are called the positive virtual (auxiliary) vector space, the negative virtual (auxiliary) vector space, respectively, and $V_{n}$ is called the (negative) proper vector space.

The inclusion $V_{n}\subset W$ induces a natural embedding of the ordinary general linear Lie algebra $gl(n)$ of $V_{n}$ into the auxiliary general linear Lie superalgebra $gl(m_{0}|m_{1}+n)$ of $W=W_{0}\oplus W_{1}$ (see, e.g. [18], [26]).

Let $A_{0}=\{\alpha_{1},\ldots,\alpha_{m_{0}}\},$ $A_{1}=\{\beta_{1},\ldots,\beta_{m_{1}}\},$ $L=\{1,2,\ldots,n\}$ denote fixed bases of $V_{m_{0}}$, $V_{m_{1}}$ and $V_{n}$, respectively; therefore $|\alpha_{s}|=0\in\mathbb{Z}_{2},$ and $|\beta_{t}|=|i|=1\in\mathbb{Z}_{2}.$

Let

[TABLE]

be the standard $\mathbb{Z}_{2}-$homogeneous basis of the Lie superalgebra $gl(m_{0}|m_{1}+n)$ provided by the elementary matrices. The elements $e_{a,b}\in gl(m_{0}|m_{1}+n)$ are $\mathbb{Z}_{2}-$homogeneous of $\mathbb{Z}_{2}-$degree $|e_{a,b}|=|a|+|b|.$

The superbracket of the Lie superalgebra $gl(m_{0}|m_{1}+n)$ has the following explicit form:

[TABLE]

$a,b,c,d\in A_{0}\cup A_{1}\cup L.$

In the following, the elements of the sets $A_{0},A_{1},L$ will be called positive virtual symbols, negative virtual symbols and negative proper symbols, respectively.

4.1.2 The supersymmetric algebra ${\mathbb{C}}[M_{m_{0}|m_{1}+n,n}]$

We regard the commutative algebra ${\mathbb{C}}[M_{n,n}]$ as a subalgebra of the “auxiliary” supersymmetric algebra

[TABLE]

generated by the ($\mathbb{Z}_{2}$-graded) variables $(\alpha_{s}|j),(\beta_{t}|j),(i|j)$, $j=1,2,\ldots,n$, where

[TABLE]

and $|(i|j)|=|i|+|j|=0,$ subject to the commutation relations:

[TABLE]

for $a,b\in\{\alpha_{1},\ldots,\alpha_{m_{0}}\}\cup\{\beta_{1},\ldots,\beta_{m_{1}}\}\cup\{1,2,\ldots,n\}.$

In plain words, all the variables commute each other, with the exception of pairs of variables $(\alpha_{s}|j),(\alpha_{t}|j)$ that skew-commute:

[TABLE]

In the standard notation of multilinear algebra, we have:

[TABLE]

where $P_{n}=(P_{n})_{1}$ denotes the trivially $\mathbb{Z}_{2}-$graded vector space with distinguished basis $\{j;\ |j|=1,\ j=1,2,\ldots,n\}.$

The algebra ${\mathbb{C}}[M_{m_{0}|m_{1}+n,n}]$ is a supersymmetric $\mathbb{Z}_{2}-$graded algebra (superalgebra), whose $\mathbb{Z}_{2}-$graduation is inherited by the natural one in the exterior algebra.

4.1.3 Left superderivations and left superpolarizations

A left superderivation $D^{\textit{l}}$ ($\mathbb{Z}_{2}-$homogeneous of degree $|D^{\textit{l}}|$) (see, e.g. [26], [18]) on ${\mathbb{C}}[M_{m_{0}|m_{1}+n,n}]$ is an element of the superalgebra $End_{\mathbb{C}}[\mathbb{C}[M_{m_{0}|m_{1}+n,n}]]$ that satisfies "Leibniz rule"

[TABLE]

for every $\mathbb{Z}_{2}-$homogeneous of degree $|\textbf{p}|$ element $\textbf{p}\in\mathbb{C}[M_{m_{0}|m_{1}+n,n}].$

Given two symbols $a,b\in A_{0}\cup A_{1}\cup L$, the left superpolarization $D^{\textit{l}}_{a,b}$ of $b$ to $a$ is the unique left superderivation of ${\mathbb{C}}[M_{m_{0}|m_{1}+n,n}]$ of $\mathbb{Z}_{2}-$degree $|D^{\textit{l}}_{a,b}|=|a|+|b|\in\mathbb{Z}_{2}$ such that

[TABLE]

Informally, we say that the operator $D^{\textit{l}}_{a,b}$ annihilates the symbol $b$ and creates the symbol $a$.

4.1.4 The superalgebra ${\mathbb{C}}[M_{m_{0}|m_{1}+n,n}]$ as a $\mathbf{U}(gl(m_{0}|m_{1}+n))$-module

Since

[TABLE]

the map

[TABLE]

is a Lie superalgebra morphism from $gl(m_{0}|m_{1}+n)$ to End_{\mathbb{C}}\big{[}\mathbb{C}[M_{m_{0}|m_{1}+n,n}]\big{]} and, hence, it uniquely defines a representation:

[TABLE]

In the following, we always regard the superalgebra $\mathbb{C}[M_{m_{0}|m_{1}+n,n}]$ as a $\mathbf{U}(gl(m_{0}|m_{1}+n))-$supermodule, with respect to the action induced by the representation $\varrho$:

[TABLE]

for every $\mathbf{p}\in{\mathbb{C}}[M_{m_{0}|m_{1}+n,n}].$

We recall that $\mathbf{U}(gl(m_{0}|m_{1}+n))-$module $\mathbb{C}[M_{m_{0}|m_{1}+n,n}]$ is a semisimple module, whose simple submodules are - up to isomorphism - Schur supermodules (see, e.g. [3], [4], [1]. For a more traditional presentation, see also [13]).

Clearly, $\mathbf{U}(gl(0|n))=\mathbf{U}(gl(n))$ is a subalgebra of $\mathbf{U}(gl(m_{0}|m_{1}+n))$ and the subalgebra $\mathbb{C}[M_{n,n}]$ is a $\mathbf{U}(gl(n))-$submodule of $\mathbb{C}[M_{m_{0}|m_{1}+n,n}]$.

4.2 The virtual algebra $Virt(m_{0}+m_{1},n)$ and the virtual

presentations of elements in $\mathbf{U}(gl(n))$

We say that a product

[TABLE]

is an irregular expression whenever there exists a right subword

[TABLE]

$i\leq m$ and a virtual symbol $\gamma\in A_{0}\cup A_{1}$ such that

[TABLE]

The meaning of an irregular expression in terms of the action of $\mathbf{U}(gl(m_{0}|m_{1}+n))$ by left superpolarization on the algebra $\mathbb{C}[M_{m_{0}|m_{1}+n,n}]$ is that there exists a virtual symbol $\gamma$ and a right subsequence in which the symbol $\gamma$ is annihilated more times than it was already created and, therefore, the action of an irregular expression on the algebra $\mathbb{C}[M_{n,n}]$ is zero.

Example 4.1**.**

Let $\gamma\in A_{0}\cup A_{1}$ and $x_{i},x_{j}\in L.$ The product

[TABLE]

is an irregular expression.

∎

Let $\mathbf{Irr}$ be the left ideal of $\mathbf{U}(gl(m_{0}|m_{1}+n))$ generated by the set of irregular expressions.

Proposition 4.2**.**

The superpolarization action of any element of $\mathbf{Irr}$ on the subalgebra $\mathbb{C}[M_{n,n}]\subset\mathbb{C}[M_{m_{0}|m_{1}+n,n}]$ - via the representation $\varrho$ - is identically zero.

Proposition 4.3**.**

([5], [2])* The sum ${\mathbf{U}}(gl(0|n))+\mathbf{Irr}$ is a direct sum of vector subspaces of $\mathbf{U}(gl(m_{0}|m_{1}+n)).$*

Proposition 4.4**.**

([5], [2])* The direct sum vector subspace $\mathbf{U}(gl(0|n))\oplus\mathbf{Irr}$ is a subalgebra of $\mathbf{U}(gl(m_{0}|m_{1}+n)).$*

The subalgebra

[TABLE]

is called the virtual algebra.

The proof of the following proposition is immediate from the definitions.

Proposition 4.5**.**

The left ideal $\mathbf{Irr}$ of ${\mathbf{U}}(gl(m_{0}|m_{1}+n))$ is a two sided ideal of $Virt(m_{0}+m_{1},n).$

The Capelli devirtualization epimorphism is the surjection

[TABLE]

with $Ker(\mathfrak{p})=\mathbf{Irr}.$

Any element in $\textbf{M}\in Virt(m_{0}+m_{1},n)$ defines an element in $\textbf{m}\in\mathbf{U}(gl(n))$ - via the map $\mathfrak{p}$ - and M is called a virtual presentation of m.

Since the map $\mathfrak{p}$ a surjection, any element $\mathbf{m}\in\mathbf{U}(gl(n))$ admits several virtual presentations. In the sequel, we even take virtual presentations as the true definition of special elements in $\mathbf{U}(gl(n)),$ and this method will turn out to be quite effective.

Recall that ${\mathbf{U}}(gl(m_{0}|m_{1}+n))$ is a Lie module with respect to the adjiont representation $Ad_{gl(m_{0}|m_{1}+n)}$. Since $gl(n)=gl(0|n)$ is a Lie subalgebra of $gl(m_{0}|m_{1}+n)$, ${\mathbf{U}}(gl(m_{0}|m_{1}+n))$ is a $gl(n)-$module with respect to the adjoint action $Ad_{gl(n)}$ of $gl(n)$.

The following results follow from the definitions.

Proposition 4.6**.**

The virtual algebra $Virt(m_{0}+m_{1},n)$ is a submodule of ${\mathbf{U}}(gl(m_{0}|m_{1}+n))$ with respect to the adjoint action $Ad_{gl(n)}$ of $gl(n)$.

Proposition 4.7**.**

The Capelli epimorphism

[TABLE]

is an $Ad_{gl(n)}-$equivariant map.

Corollary 4.8**.**

The isomorphism $\mathfrak{p}$ maps any $Ad_{gl(n)}-$invariant element $\mathbf{m}\in Virt(m_{0}+m_{1},n)$ to a central element of $\mathbf{U}(gl(n))$.

Balanced monomials are elements of the algebra ${\mathbf{U}}(gl(m_{0}|m_{1}+n))$ of the form:

• –

$e_{{i_{1}},\gamma_{p_{1}}}\cdots e_{{i_{k}},\gamma_{p_{k}}}\cdot e_{\gamma_{p_{1}},{j_{1}}}\cdots e_{\gamma_{p_{k}},{j_{k}}},$

• –

$e_{{i_{1}},\theta_{q_{1}}}\cdots e_{{i_{k}},\theta_{q_{k}}}\cdot e_{\theta_{q_{1}},\gamma_{p_{1}}}\cdots e_{\theta_{q_{k}},\gamma_{p_{k}}}\cdot e_{\gamma_{p_{1}},{j_{1}}}\cdots e_{\gamma_{p_{k}},{j_{k}}},$

• –

and so on,

where $i_{1},\ldots,i_{k},j_{1},\ldots,j_{k}\in L,$ i.e., the $i_{1},\ldots,i_{k},j_{1},\ldots,j_{k}$ are $k$ proper (negative) symbols, and the $\gamma_{p_{1}},\ldots,\gamma_{p_{k}},\ldots,\theta_{q_{1}},\ldots,\theta_{q_{k}},\ldots$ are virtual symbols. In plain words, a balanced monomial is product of two or more factors where the rightmost one annihilates (by superpolarization) the $k$ proper symbols $j_{1},\ldots,j_{k}$ and creates (by superpolarization) some virtual symbols; the leftmost one annihilates all the virtual symbols and creates the $k$ proper symbols $i_{1},\ldots,i_{k}$; between these two factors, there might be further factors that annihilate and create virtual symbols only.

Proposition 4.9**.**

([3], [4], [1], [2])* Every balanced monomial belongs to $Virt(m_{0}+m_{1},n)$. Hence, the Capelli epimorphism $\mathfrak{p}$ maps balanced monomials to elements of $\mathbf{U}(gl(n)).$*

Let $S$ and $T$ be the Young tableaux

[TABLE]

To the pair $(S,T)$, we associate the bitableau monomial:

[TABLE]

in ${\mathbf{U}}(gl(m_{0}|m_{1}+n)).$

Let $\beta_{1},\ldots,\beta_{\lambda_{1}}\in A_{1}$, $\alpha_{1},\ldots,\alpha_{p}\in A_{0}$ be sets of negative and positive virtual symbols, respectively. Set

[TABLE]

.

The tableaux $D_{\lambda}$ and $C_{\lambda}$ are called the virtual Deruyts and Coderuyts tableaux of shape $\lambda,$ respectively.

Given a pair of Young tableaux $S,T$ of the same shape $\lambda$ on the proper alphabet $L$, consider the elements

[TABLE]

[TABLE]

[TABLE]

Since elements (13), (14) and (15) are balanced monomials in ${\mathbf{U}}(gl(m_{0}|m_{1}+n))$, they belong to the subalgebra $Virt(m_{0}+m_{1},n)$.

We set

[TABLE]

and call the element $[S|T]$ a Capelli bitableau [5], [6].

We set

[TABLE]

and call the element $[S|T]^{*}$ a *Capelli -bitableau [5], [6].

We set

[TABLE]

and call the element [S|\framebox{T}] a right Young-Capelli bitableau [4].

5 The bitableaux correspondence maps $\mathcal{B}$ and $\mathcal{B}^{*}$

and the Koszul map $\mathcal{K}$

Theorem 5.1**.**

The bitableaux correspondence map

[TABLE]

uniquely extends to a linear map

[TABLE]

Proof.

We recall that bitableaux and Capelli bitableaux satisfy the same (determinantal) straightening laws in ${\mathbb{C}}[M_{n,n}]$ and $\mathbf{U}(gl(n))$, respectively ([5], Proposition $7$). The straightening laws imply that standard (determinantal) bitableaux span ${\mathbb{C}}[M_{n,n}]$ (see, e.g. [16], [14], [15]); furhermore, standard bitableaux are linearly independent. Then, the map $\mathcal{B}$ is a uniquely defined linear operator. ∎

Theorem 5.2**.**

*The -bitableaux correspondence map

[TABLE]

uniquely extends to a linear map

[TABLE]

Proof.

The proof is essentially the same as the proof of Theorem 5.1, just by replacing the determinantal straightening laws with the permanental straightening laws, and standard (determinantal) bitableaux with costandard (permanental) bitableaux. Notice that both arguments are special cases of the superalgebraic version of the straightening laws and of the standard basis theorem ([17], [1]). ∎

Given $i,j=1,2,\ldots,n$, let

[TABLE]

be the linear operator

[TABLE]

Proposition 5.3**.**

We have:

[TABLE]

∎

By the universal property of $\mathbf{U}(gl(n))$, Proposition 5.3 implies

Proposition 5.4**.**

The map

[TABLE]

defines an associative algebra morphism

[TABLE]

∎

Let $\varepsilon_{1}$ be the linear map evaluation at $1$

[TABLE]

[TABLE]

The Koszul map [19] is the (linear) composition map

[TABLE]

[TABLE]

Proposition 5.5**.**

We have:

$\mathcal{K}(e_{i_{1}j_{1}}e_{i_{2}j_{2}}\cdots e_{i_{h}j_{h}})=\rho_{i_{1}j_{1}}\rho_{i_{2}j_{2}}\cdots\rho_{i_{h}j_{h}}(1),$*  $e_{i_{p}j_{p}}\in gl(n)$, $p=12,\ldots,h$.* 2. 2.

$\mathcal{K}(e_{ij}\mathbf{P})=\rho_{ij}(\mathcal{K}(\mathbf{P})),$*  for every $\mathbf{P}\in\mathbf{U}(gl(n))$, $\ e_{ij}\in gl(n)$.*

∎

6 Expansion formulae for

column Capelli bitableaux and *column Capelli -bitableaux

Consider the column Capelli bitableau

[TABLE]

(where $\alpha_{1},\ldots,\alpha_{h}$ are arbitrary distict positive virtual symbols) and the *column Capelli -bitableau

[TABLE]

(where $\beta_{1},\ldots,\beta_{h}$ are arbitrary distict negative virtual symbols).

Remember that the proper symbols $i_{1},\ldots,i_{h},\ j_{1},\ldots,j_{h}\in L=\{1,\ 2,\ \ldots,\ n\}$ are assumed to be negative.

From the definitions, it follows

[TABLE]

From the definitions, we infer

Proposition 6.1**.**

*Column Capelli bitableaux and column Capelli -bitableaux are row-commutative as elements of $\mathbf{U}(gl(n))$:

[TABLE] 2. 2.

[TABLE]

∎

We provide two basic expansion formulae that describe the effect of picking out (on the left hand side) the first row of column Capelli bitableaux and column Capelli *-bitableaux. These formulae play a crucial role in the theory of the Koszul map $\mathcal{K}$, and provide a simple way to compute the actual forms of column Capelli bitableaux and column Capelli *-bitableaux as elements of $\mathbf{U}(gl(n))$.

Proposition 6.2**.**

We have:

[TABLE] 2. 2.

[TABLE]

Proof.

By definition,

[TABLE]

[TABLE]

Notice that

[TABLE]

[TABLE]

as elements of the algebra ${\mathbf{U}}(gl(m_{0}|m_{1}+n)).$

Therefore, the summand

[TABLE]

equals

[TABLE]

By repeating the above procedure of moving left the element $e_{\alpha_{1},{j_{1}}}$ - using the commutator identities in ${\mathbf{U}}(gl(m_{0}|m_{1}+n))$ - we finally get

[TABLE]

[TABLE]

Notice that the summand

[TABLE]

equals

[TABLE]

[TABLE]

as elements of the algebra ${\mathbf{U}}(gl(m_{0}|m_{1}+n)).$

Hence

[TABLE]

equals

[TABLE]

Furthermore

[TABLE]

[TABLE]

Since column Capelli bitableaux are row-commutative, by setting $k=h-i$ we proved the first expansion identity. The second expansion identity can be proved in a similar way. ∎

Example 6.3**.**

[TABLE]

∎

Notice that, for $h=1$, $[i|j]=[i|j]^{*}=e_{ij}$. Then, from Proposition 6.2, it follows

Corollary 6.4**.**

The family of column Capelli bitableaux (-bitableaux) is a system of linear generators of $\mathbf{U}(gl(n))$.*

7 Main results

Proposition 7.1**.**

[TABLE]

Proof.

[TABLE]

∎

Example 7.2**.**

Consider the column Capelli bitableau

[TABLE]

We have

[TABLE]

∎

Proposition 7.3**.**

[TABLE]

Proof.

[TABLE]

[TABLE]

∎

Notice that Theorem 5.1 specializes to

[TABLE]

and, Theorem 5.2 specializes to

[TABLE]

Theorem 7.4**.**

We have:

$\mathcal{B}=\mathcal{K}^{-1}$, 2. 2.

$\mathcal{B^{*}}=\mathcal{K}^{-1}$, 3. 3.

$\mathcal{B},\ \mathcal{B^{*}},\ \mathcal{K}$* are linear isomorphisms,* 4. 4.

$\mathcal{B}=\mathcal{B^{*}}$.

Proof.

From Corollary 6.4 and Eqs. (17) and (18), it follows that the operators $\mathcal{B}$ and $\mathcal{B^{*}}$ are surjective. Since column bitableaux span ${\mathbb{C}}[M_{n,n}]$, Propositions 7.1 and 7.3 imply that $\mathcal{B}$ and $\mathcal{B^{*}}$ are injective and $\mathcal{B}=\mathcal{K}^{-1}$ and $\mathcal{B^{*}}=\mathcal{K}^{-1}$. Then $\mathcal{B}=\mathcal{B^{*}}$. ∎

By combining Theorems 5.1 and 5.2 with Theorem 7.4, it follows

Corollary 7.5**.**

We have:

• –

$\mathcal{K}:[S|T]\mapsto(S|T),$**

• –

$\mathcal{K}:[S|T]^{*}\mapsto(S|T)^{*}.$**

∎

The Koszul isomorphism $\mathcal{K}$ well-behaves with respect to right symmetrized bitableaux and right Young-Capelli bitableaux.

Proposition 7.6**.**

We have:

[TABLE]

Proof.

We notice that

[TABLE]

where the sum is extended over all $\overline{T}$ column permuted of $T$ (hence, repeated entries in a column give rise to multiplicities). The proof of the first equality easily follows from the definition, by applying the commutator identities in the superalgebra ${\mathbf{U}}(gl(m_{0}|m_{1}+n))$. The second equality is the definition of the right symmetrized bitableau (S|\framebox{T})), Eq. (10). ∎

From Proposition 2.2, Corollary 7.5 and Proposition 7.6, it follows

Corollary 7.7**.**

*The sets of standard Capelli bitableaux, of costandard Capelli -bitableaux and of standard Young-Capelli bitableaux:

• –

\Big{\{}[S|T];\ sh(S)=sh(T)=\lambda,\ \lambda_{1}\leq n,\ S,T\ standard\Big{\}},

• –

\Big{\{}[U|V]^{*};\ sh(U)=sh(V)=\mu,\ \widetilde{\mu_{1}}\leq n,\ U,V\ costandard\Big{\}},

• –

\Big{\{}[S|\framebox{T}];\ \ sh(S)=sh(T)=\lambda,\ \lambda_{1}\leq n,\ S,T\ standard\Big{\}}**

are linear bases of $\mathbf{U}(gl(n))$.

Furthermore, we have

Theorem 7.8**.**

The Koszul isomorphism $\mathcal{K}$ is equivariant with respect to the adjoint representations \big{(}Ad_{gl(n)},\mathbf{U}(gl(n))\big{)} and \big{(}ad_{gl(n)},{\mathbb{C}}[M_{n,n}]\big{)}.

Proof.

We recall that the action of $e_{hk}\in gl(n)$ on $\mathbf{U}(gl(n))$ through the adjoint representation $Ad_{gl(n)}$ is implemented by the derivation $T_{hk}$ such that T_{hk}\big{(}e_{st}\big{)}=\delta_{ks}e_{it}-\delta_{ht}e_{sj}. From the definition of column Capelli bitableau and Proposition 4.7, we infer

[TABLE]

We recall that the action of $e_{hk}$ on ${\mathbb{C}}[M_{n,n}]$ through the adjoint representation $ad_{gl(n)}$ is implemented by the derivation $D^{\textit{l}}_{hk}-D^{\textit{r}}_{kh}$. Then

[TABLE]

Since column Capelli bitableaux span $\mathbf{U}(gl(n))$ and column bitableaux span ${\mathbb{C}}[M_{n,n}]$, the assertion follows from Proposition 7.1. ∎

Since

[TABLE]

the preceding Theorem implies:

Corollary 7.9**.**

We have

[TABLE]

In the left representation \big{(}\rho^{\textit{l}},{\mathbb{C}}[M_{n,n}]\big{)} (i.e. $\rho^{\textit{l}}:e_{ij}\mapsto D^{\textit{l}}_{ij}$), standard Young-Capelli bitableaux [S|\framebox{T}], $sh(S)=sh(T)=\lambda\vdash k$, act on right symmetrized bitableaux (U|\framebox{V}), $sh(U)=sh(V)=\mu\vdash h$, in a quite remarkable way.

Proposition 7.10**.**

[4]* We have:*

• –

If $h<k$, the action is zero.

• –

If $h=k$ and $\lambda\neq\mu$, the action is zero.

• –

If $h=k$ and $\lambda=\mu$, the action is nondegenerate triangular (with respect to a suitable linear order on standard tableaux of the same shape).

For details and proof, see [1] Theorem $10.1$. Clearly, similar results hold for the right and the adjoint representations.

8 Laplace expansions

8.1 Laplace expansions in ${\mathbb{C}}[M_{n,n}]$

Recall that

[TABLE]

and, therefore, the biproduct $(i_{1}i_{2}\cdots i_{h}|j_{1}j_{2}\cdots j_{h})\in{\mathbb{C}}[M_{n,n}]$ expands into column bitableaux as follows:

[TABLE]

Notice that, in the passage from monomials to column bitableaux, the sign $(-1)^{h\choose 2}$ disappears.

Recall that

[TABLE]

and, therefore, the *-biproduct $(i_{1}i_{2}\cdots i_{h}|j_{1}j_{2}\cdots j_{h})^{*}\in{\mathbb{C}}[M_{n,n}]$ expands into column *-bitableaux as follows:

[TABLE]

The preceding arguments extend to bitableaux and to *-bitableaux of any shape $\lambda,\ \lambda_{1}\leq n.$ Given the Young tableaux

[TABLE]

From a simple sign computation, it follows

Proposition 8.1**.**

[TABLE]

[TABLE]

where the multiple sums range over all permutations $\sigma_{1}\in\mathbf{S}_{\lambda_{1}},\ldots,\sigma_{m}\in\mathbf{S}_{\lambda_{m}}.$

Notice that, in the expansions with respect to column bitableax, only the signs of permutations will remain.

Similarly, we have

Proposition 8.2**.**

[TABLE]

[TABLE]

8.2 Laplace expansions in $\mathbf{U}(gl(n))$

Let $S$ and $T$ be the Young tableaux

[TABLE]

Propositions 8.1 and 8.2 and Theorems 5.1 and 5.2 imply to the following Laplace expansions of Capelli bitableaux into column Capelli bitableaux and of Capelli *-bitableaux into column Capelli *-bitableaux.

Corollary 8.3**.**

We have

[TABLE]

[TABLE]

Corollary 8.4**.**

We have

[TABLE]

[TABLE]

By combining the expansions of Corollaries 8.3 and 8.4 with the results of Proposition 6.2, one gets explicit expansions of Capelli bitableaux and of Capelli *-bitableaux as elements of $\mathbf{U}(gl(n))$.

Example 8.5**.**

The Capelli bitableau (of shape $\lambda=(2,2)$)

[TABLE]

equals

[TABLE]

where

[TABLE]

This example can be used to enlighten the difference between the PBW Theorem and Theorem 7.4.

The PBW Theorem establishes an isomorphism $\phi$ from the graded algebra

[TABLE]

*associated to the filtered algebra $\mathbf{U}(gl(n))$ to the algebra $\mathbf{Sym}(gl(n))\cong{\mathbb{C}}[M_{n,n}]$. Clearly, the isomorphism $\phi$ maps the projection to $Gr\left[\mathbf{U}(gl(n))\right]$ of the Capelli bitableau (19)

• as an element of the quotient space $\frac{\mathbf{U}^{(4)}(gl(4))}{\mathbf{U}^{(3)}(gl(4))}$ - to the product determinants*

[TABLE]

*The Koszul isomorphism $\mathcal{K}$ (injectively) maps the Capelli bitableau (19)

• as an element of $\mathbf{U}(gl(4))$ - to the product of determinants (20). Similarly, the isomorphism $\mathcal{K}$ maps the Capelli -bitableau

[TABLE]

to the product permanents

[TABLE]

∎

In the following, we will discuss some implications of Corollary 7.9.

Proposition 8.6**.**

(Koszul [19])* Consider the row Capelli bitableau*

[TABLE]

We have:

[TABLE]

the **Capelli column determinant111The symbol $\mathbf{cdet}$ denotes the column determinat of a matrix $A=[a_{ij}]$ with noncommutative entries: $\mathbf{cdet}(A)=\sum_{\sigma}\ (-1)^{|\sigma|}\ a_{\sigma(1),1}a_{\sigma(2),2}\cdots a_{\sigma(n),n}.$* in $\mathbf{U}(gl(n))$.* 2. 2.

[TABLE]

Proof.

We have

[TABLE]

from Proposition 6.2.

By iterating the same argument,

[TABLE]

the Capelli column determinant in $\mathbf{U}(gl(n))$.

Then

[TABLE]

that equals

[TABLE]

by Corollary 7.5. ∎

In the enveloping algebra $\mathbf{U}(gl(n))$, given any integer $k=1,2,\ldots,n,$ consider the $k-$th Capelli element:

[TABLE]

By the same argument of Proposition 8.6,

[TABLE]

and the operator $\mathcal{K}$ maps $\mathbf{H}_{k}(n)$ to the polynomial

[TABLE]

Notice that the polynomials $\mathbf{h}_{k}(n)$’s appear as coefficients (in ${\mathbb{C}}[M_{n,n}]$) of the characteristic polynomial:

[TABLE]

Clearly, $\mathbf{h}_{k}(n)$ is $ad_{gl(n)}-$invariant in ${\mathbb{C}}[M_{n,n}]$ and, therefore, $\mathbf{H}_{k}(n)$ is a central element of the enveloping algebra $\mathbf{U}(gl(n))$.

In passing we recall Capelli’s Theorem ([9] and [10], see also [6]):

Proposition 8.7**.**

[TABLE]

Moreover, the $\mathbf{H}_{k}(n)$’s are algebraically independent.

In general, given a partition $\lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{p})$, $\lambda_{1}\leq n$, consider the sum of Capelli bitableaux

[TABLE]

where the sum is extended to all row-increasing tableaux $S$, $sh(S)=\lambda$ (the $\mathbf{K}_{\lambda}(n)$’s are called shaped Capelli elements in [7]). Notice that the elements $\mathbf{K}_{\lambda}(n)$ are radically different from the elements $\mathbf{H}_{\lambda}(n)=\mathbf{H}_{\lambda_{1}}(n)\cdots\mathbf{H}_{\lambda_{p}}(n)$.

From Corollary 7.5, Eqs. (3) and (4) and row skew-symmetry of bitableaux, we infer

Proposition 8.8**.**

We have

[TABLE]

Hence, the elements $\mathbf{K}_{\lambda}(n)$ are central. By Corollary 7.9, the following statements are equivalent:

• –

The $\mathbf{K}_{\lambda}(n)-$basis theorem for $\boldsymbol{\zeta}(n)$ [7]:

Proposition 8.9**.**

The set

[TABLE]

is a linear basis of $\boldsymbol{\zeta}(n)$.

Notice that the elements $\mathbf{K}_{\lambda}(n)$ are radically different from the quantum immanants of [21], [22] and [8].

• –

The well-known theorem for the algebra of invariants ${\mathbb{C}}[M_{n,n}]^{ad_{gl(n)}}$:

Proposition 8.10**.**

[TABLE]

Moreover, the $\mathbf{h}_{k}(n)$’s are algebraically independent.

Proposition 8.10 is usually stated in terms of the algebra ${\mathbb{C}}[M_{n,n}]^{GL(n)}={\mathbb{C}}[M_{n,n}]^{ad_{gl(n)}}$, where ${\mathbb{C}}[M_{n,n}]^{GL(n)}$ is the subalgebra of invariants with respect to the conjugation action of the general linear group ${GL(n)}$ on ${\mathbb{C}}[M_{n,n}]$ (see, e.g. [20], [14], [24]).