Quantum affine wreath algebras

TL;DR

This paper introduces quantum affine wreath algebras, a new class of algebras that generalize affine Hecke algebras and affine wreath algebras through deformation techniques, and explores their structural properties.

Contribution

It defines quantum affine wreath algebras associated with symmetric algebras and investigates their structure and cyclotomic quotients.

Findings

Established the structure theory of quantum affine wreath algebras

Analyzed the properties of their cyclotomic quotients

Connected these algebras to existing algebraic frameworks

Abstract

To each symmetric algebra we associate a family of algebras that we call quantum affine wreath algebras. These can be viewed both as symmetric algebra deformations of affine Hecke algebras of type $A$ and as quantum deformations of affine wreath algebras. We study the structure theory of these new algebras and their natural cyclotomic quotients.

Equations

$$A\to\operatorname{Hom}_{\Bbbk}(A,\Bbbk),\quad a\mapsto\big{(}b\mapsto\mathrm{tr}(ab)\big{)},$$

$$\mathrm{tr}(ab)=(-1{)}^{\bar{a}\bar{b}}\mathrm{tr}(ba),a,b\in A.$$

$$\mathrm{tr}(a^{\vee }b)={\delta }_{a,b},a,b\in B.$$

$$(b^{\vee }{)}^{\vee }=(-1{)}^{\bar{b}}b,b\in B.$$

$$a_i=1^{\otimes (i-1)}\otimes a\otimes 1^{\otimes (n-i)}\in A^{\otimes n}.$$

$$A^{\otimes n}\star ⟨T_i:1\le i\le n-1⟩,$$

$$T_i{T}_j$$

$$T_i{T}_{i+1}{T}_i$$

$$T_i^2$$

$$T_i\mathbf{a}$$

$$t_{i,j}:=\sum _{b\in B}{b}_i{b}_j^{\vee },1\le i,j\le n-1,$$

$$T_w=T_{i_1}{T}_{i_2}\cdots T_{i_k},$$

$$t_{i,j}=t_{j,i},1\le i,j\le n.$$

$$T_i{t}_{j,k}=t_{s_i(j),s_i(k)}{T}_i,1\le i\le n-1,1\le j,k\le n.$$

$$T_i{t}_{i,i+1}=t_{i,i+1}{T}_i,1\le i\le n-1.$$

$$T_i-T_i^{-1}=z{t}_{i,i+1},1\le i\le n-1.$$

$$\mathbf{a}{t}_{j,k}=t_{j,k}{s}_{j,k}(\mathbf{a}),\mathbf{a}\in A^{\otimes n},1\le i,j,k\le n,$$

$$\mathbb{k}[X_1^{\pm 1},\dots ,X_n^{\pm 1}]\star H_n(A,z),$$

$$T_i{X}_j$$

$$T_i{X}_i{T}_i$$

$$X_i\mathbf{a}$$

$$X_i^{\pm 1}\mapsto X_{n+1-i}^{\pm 1},a_i\mapsto a_{n+1-i},T_j\mapsto -T_{n-j}^{-1}=-T_{n-j}+z{t}_{n-j,n-j+1},$$

$$X_i^{\pm 1}\mapsto X_i^{\pm 1},\mathbf{a}\mapsto {\xi }^{\otimes n}(\mathbf{a}),T_j\mapsto T_j,$$

$$\hat{\tau }:H_n^{\text{aff}}(A,z)\to H_n^{\text{aff}}(A,z{)}^{\mathrm{op}},X_i^{\pm 1}\mapsto X_i^{\pm 1},\mathbf{a}\mapsto {\tau }^{\otimes n}(\mathbf{a}),T_j\mapsto T_j,$$

$$Z(A):=\left\{a\in A:ab=(-1{)}^{\bar{a}\bar{b}}ba\text{ for all }b\in A\right\}.$$

$$\mathbf{a}\mapsto \mathbf{a},T_i\mapsto T_i,X_j\mapsto a_j{X}_j,X_j^{-1}\mapsto a_j^{-1}{X}_j^{-1},$$

$$P_n=\mathbb{k}[X_1^{\pm 1},\dots ,X_n^{\pm 1}]\text{and}{P}_n(A)=A^{\otimes n}\otimes P_n\text{(tensor product of algebras)}.$$

$${}^w(\mathbf{a}\otimes p)=\mathbf{a}\otimes w(p).$$

$${\Delta }_i:P_n(A)\to P_n(A),{\Delta }_i(f)=\frac{f-{}^{s_i}f}{1-X_i{X}_{i+1}^{-1}}.$$

$${\Delta }_i(fg)={\Delta }_i(f)g+{}^{s_i}f{\Delta }_i(g),f,g\in P_n(A).$$

$${\Delta }_i(fg)=f{\Delta }_i(g)\text{if}{}^{s_i}f=f.$$

$${}^{s_{i}}\Delta_{i}(f)=X_{i}X_{i+1}^{-1}\Delta_{i}(f),\quad\Delta_{i}\big{(}{}^{s_{i}}f\big{)}=-\Delta_{i}(f),\quad\Delta_{i}\big{(}{}^{s_{j}}f\big{)}={}^{s_{j}}\Delta_{i}(f),$$

$$\displaystyle s_{i}\big{(}\Delta_{i}(f)\big{)}=-X_{i}X_{i+1}^{-1}\Delta_{i}\big{(}s_{i}(f)\big{)},\quad s_{i}\big{(}\Delta_{j}(f)\big{)}=\Delta_{j}\big{(}s_{i}(f)\big{)},$$

Full text

1. Introduction

Affine Hecke algebras and their degenerate versions are fundamental in the study of Lie algebras and quantum groups. Affine wreath algebras, whose systematic study was undertaken in [Sav20], provide a unifying and generalizing framework for various modified versions of degenerate affine Hecke algebras (of type $A$) appearing in the literature.111The terminology affine wreath product algebras was used in [Sav20]. We drop the word “product” in the current paper for simplicity. (Certain cases of these algebras were also considered in [KM19].) Affine wreath algebras also occur naturally as endomorphism algebras in the Frobenius Heisenberg categories of [Sav19, RS17]. It is natural to ask if such a general approach exists in the quantum (i.e. non-degenerate) setting. The purpose of the current paper is to answer this question in the affirmative.

Fix a commutative ground ring $\Bbbk$ and $z\in\Bbbk$. To any symmetric superalgebra $A$, we associate a quantum wreath algebra $H_{n}(A,z)$. The superalgebra $H_{n}(A,z)$ can be viewed as a $z$-deformation of the wreath algebra $A^{\otimes n}\rtimes S_{n}$, in the sense that $H_{n}(A,0)=A^{\otimes n}\rtimes S_{n}$. Simultaneously, $H_{n}(A,z)$ can be thought of as an $A$-deformation of the Iwahori–Hecke algebra of type $A$. In particular, taking $A=\Bbbk=\mathbb{C}[q,q^{-1}]$ and $z=q-q^{-1}$ recovers the Iwahori–Hecke algebra. We then define an affine version, the quantum affine wreath algebra $H_{n}^{\textup{aff}}(A,z)$. Again, this superalgebra can be viewed simultaneously as a $z$-deformation of the affine wreath algebras of [Sav20] and as an $A$-deformation of the affine Hecke algebra of type $A$.

The quantum affine wreath algebras defined in the current paper unify and generalize existing analogs of affine Hecke algebras. In particular, we have the following:

1. (a)

When $A=\mathbb{C}$, the affine wreath algebra is the degenerate affine Hecke algebra of type $A$. As noted above, when $A=\mathbb{C}[q,q^{-1}]$ and $z=q-q^{-1}$, the quantum affine wreath algebra is the affine Hecke algebra of type $A$. 2. (b)

When $A$ is the group algebra of a finite group $G$, the affine wreath algebra is the wreath Hecke algebra of Wan and Wang [WW08]. When $G$ is a finite cyclic group, the quantum (affine) wreath algebra is the (affine) Yokonuma–Hecke algebra. (See Examples 2.4 and 2.7 for details.) For more general groups, the quantum affine wreath algebra seems to be new. 3. (c)

When $A$ is a certain skew-zigzag algebra (see [HK01, §3] and [Cou16, §5]), the corresponding affine wreath algebras appear in the endomorphism algebras of the categories constructed in [CL12] to study Heisenberg categorification and the geometry of Hilbert schemes. They were then also considered in [KM19], where they were related to imaginary strata for quiver Hecke algebras (also known as KLR algebras).

For this choice of $A$, the quantum affine wreath algebras of the current paper yield natural $z$-deformations of these affine zigzag algebras. These deformations seem to be new.

Despite their high level of generality, one can deduce a great deal of the structure of quantum affine wreath algebras. Specializing the symmetric superalgebra $A$ then recovers known results in some cases and new results in others. In addition, just as the affine wreath algebras appear as endomorphism algebras in the Frobenius Heisenberg categories of [Sav19, RS17], the quantum affine wreath algebras defined in the current paper appear as endomorphism algebras in the quantum Frobenius Heisenberg categories of [BSW]. In fact, this is one of the main motivations of the current paper. The quantum Frobenius Heisenberg category acts on categories of modules for the quantum cyclotomic wreath algebras introduced here. This action generalizes the action of the quantum Heisenberg category of [BSW18] (see also [LS13]) on categories of modules for cyclotomic Hecke algebras.

We now give an overview of the main results of the current paper. We define the quantum (affine) wreath algebras in Section 2 and discuss some natural symmetries. In Section 3 we examine the structure theory of these algebras. We first introduce natural Demazure operators which are useful in computations. We then describe an explicit basis of $H_{n}^{\textup{aff}}(A,z)$ in Theorem 3.10, and the center of $H_{n}^{\textup{aff}}(A,z)$ in Theorem 3.16. Finally, we define natural Jucys–Murphy elements in Section 3.4 and give a Mackey Theorem for $H_{n}^{\textup{aff}}(A,z)$ in Theorem 3.20. In Section 4 we turn our attention to cyclotomic quotients. We define the quantum cyclotomic wreath algebra $H_{n}^{f}(A,z)$ associated to a monic polynomial $f$ with coefficients in the even part of the center $Z(A)$ of $A$. These quotients are analogues of cyclotomic Hecke algebras. We prove a basis theorem (Theorem 4.10) for these quotients and a cyclotomic Mackey Theorem (Theorem 4.14). Finally, we prove that the quantum cyclotomic wreath algebras are symmetric algebras and that $H_{n+1}^{f}(A,z)$ is a Frobenius extension of $H_{n}^{f}(A,z)$.

We expect that most of the results of the current paper can be generalized to the setting where $A$ is a Frobenius superalgebra, instead of a symmetric superalgebra (see 2.8). This more general setting was treated in the degenerate case in [Sav20] since the choice of $A$ to be the Clifford superalgebra, which is not symmetric in the super sense, yielded the affine Sergeev algebra (also called the degenerate affine Hecke–Clifford superalgebra). However, in the quantum setting of the current paper we choose to focus on the case where $A$ is symmetric for simplicity. In fact, the Clifford case is more naturally treated by considering an odd affinization of the quantum wreath algebra. This will be explored in future work.

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Acknowledgements

The first author was supported in this research by a Grant-in-Aid of Research and a Summer Faculty Fellowship from Indiana University Northwest. This research of the second author was supported by Discovery Grant RGPIN-2017-03854 from the Natural Sciences and Engineering Research Council of Canada. We thank J. Brundan for helpful conversations.

2. Definitions

In this section we introduce our main objects of study. Throughout the document, we fix a commutative ground ring $\Bbbk$ of characteristic not equal to two. (This assumption on the characteristic is not needed if one works in the non-super setting.) We also fix an element $z\in\Bbbk$. All tensor products and algebras are over $\Bbbk$ unless otherwise specified. In addition, all algebras and modules are associative superalgebras and supermodules. We drop the prefix “super” for simplicity. For a homogeneous element $a$, we use the notation $\bar{a}$ to denote its parity. We use $\mathbb{N}$ to denote the set of nonnegative integers.

2.1. Quantum wreath algebras

Fix a symmetric algebra $A$ with parity-preserving linear supersymmetric trace map $\mathrm{tr}\colon A\to\Bbbk$. (Here we consider $\Bbbk$ to live in parity zero.) In other words, the map

[TABLE]

is a parity-preserving isomorphism of $\Bbbk$-modules and, for homogeneous elements $a,b$,

[TABLE]

We will assume that $A$ is free as a $\Bbbk$-module. So we have a basis $B$ with dual basis $\{b^{\vee}:b\in B\}$ defined by

[TABLE]

It follows from the supersymmetry of the trace that

[TABLE]

Fix $n\in\mathbb{Z}_{>0}$. For $a\in A$ and $1\leq i\leq n$, we define

[TABLE]

Definition 2.1** (Quantum wreath algebra).**

For $n\in\mathbb{N}$, $n\geq 2$, we define the quantum wreath algebra (or Frobenius Hecke algebra) $H_{n}(A,z)$ to be the free product

[TABLE]

(here the angled brackets mean the free associative algebra on the given generators) modulo the relations

[TABLE]

where

[TABLE]

and $s_{i}(\mathbf{a})$ denotes the action of the simple transposition $s_{i}$ on $\mathbf{a}$ by superpermutation of the factors. It is straightforward to verify that $t_{i,j}$ does not depend on the choice of basis $B$. We adopt the conventions that $H_{1}(A,z):=A$ and $H_{0}(A,z):=\Bbbk$.

For $w\in S_{n}$, we define

[TABLE]

where $w=s_{i_{1}}s_{i_{2}}\dotsm s_{i_{k}}$ is a reduced decomposition. Since the generators $T_{i}$ satisfy the braid relations 2.2 and 2.3, this definition is independent of the choice of reduced decomposition.

Remark 2.2**.**

In the degenerate case, it was shown in [Sav20, Lem. 3.2] that affine wreath algebras depend, up to isomorphism, only on the underlying algebra $A$, and not on the trace map. However, in the quantum setting of the current paper, there do not seem to be obvious isomorphisms between quantum affine wreath algebras corresponding to the same algebra, but with different trace maps.

Example 2.3** (Iwahori–Hecke algebras).**

If $A=\Bbbk=\mathbb{C}[q,q^{-1}]$ and $z=q-q^{-1}$, then $H_{n}(A,z)$ is the Iwahori–Hecke algebra of type $A_{n-1}$.

Example 2.4** (Yokonuma–Hecke algebras).**

Let $C_{d}$ be a cyclic group of order $d$. If $\Bbbk=\mathbb{C}[q,q^{-1}]$, $z=(q-q^{-1})/d$, and $A=\Bbbk C_{d}$, with trace map given by projection onto the identity element of the group, then $H_{n}(\Bbbk C_{d},z)$ is the Yokonuma–Hecke algebra (see [CPd14, §2.1]).

It follows from 2.1 that

[TABLE]

Then, by 2.5, we have

[TABLE]

In particular,

[TABLE]

It then follows from 2.4 that the $T_{i}$ are invertible and we have a Frobenius skein relation:

[TABLE]

We also have

[TABLE]

where $s_{j,k}$ is the transposition of $j$ and $k$. For this reason, we call the $t_{i,j}$ teleporters. (In the string diagram formalism for monoidal categories, 2.7 corresponds to tokens teleporting between strands. See [Sav19, §2.1].)

2.2. Quantum affine wreath algebras

Definition 2.5** (Quantum affine wreath algebra).**

For $n\in\mathbb{N}$, $n\geq 1$, we define the quantum affine wreath algebra (or affine Frobenius Hecke algebra) $H_{n}^{\textup{aff}}(A,z)$ to be the free product of algebras

[TABLE]

modulo the relations

[TABLE]

We define $H_{n,+}^{\textup{aff}}(A,z)$ to be the subalgebra of $H_{n}^{\textup{aff}}(A,z)$ generated by $H_{n}(A,z)$ together with $\Bbbk[X_{1},\dotsc,X_{n}]$ (no inverses). We adopt the convention that $H_{0}^{\textup{aff}}(A,z)=H_{0,+}^{\textup{aff}}(A,z):=\Bbbk$.

Example 2.6** (Affine Hecke algebras).**

If $A=\Bbbk=\mathbb{C}[q,q^{-1}]$ and $z=q-q^{-1}$, then $H_{n}^{\textup{aff}}(A,z)$ is the affine Hecke algebra of type $A_{n-1}$.

Example 2.7** (Affine Yokonuma–Hecke algebras).**

In the setting of Example 2.4, $H_{n}^{\textup{aff}}(\Bbbk C_{d},z)$ is the affine Yokonuma–Hecke algebra (see [CPd14, §3.1]).

Remark 2.8**.**

One can work in the more general setting where $A$ is a Frobenius algebra. In general, there exists a Nakayama automorphism $\psi\colon A\to A$ such that $\mathrm{tr}(ab)=(-1)^{\bar{a}\bar{b}}\mathrm{tr}(b\psi(a))$ for all $a,b\in A$. Then we modify the relation 2.10 to be $\mathbf{a}X_{i}=X_{i}\psi_{i}(\mathbf{a})$, where $\psi_{i}=1^{\otimes(i-1)}\otimes\psi\otimes 1^{\otimes(n-i)}$. In the current paper we focus on the symmetric case, where $\psi=1$, for simplicity. However, it is this more general setting that motivates our use of the word “Frobenius” in some of our terminology.

2.3. Symmetries

It is straightforward to verify that we have an algebra automorphism of $H_{n}^{\textup{aff}}(A,z)$ given by

[TABLE]

for all $1\leq i\leq n$, $1\leq j\leq n-1$, $a\in A$.

Any algebra automorphism $\xi\colon A\to A$ preserving the trace (i.e. $\mathrm{tr}\circ\xi=\mathrm{tr}$) induces an algebra automorphism of $H_{n}^{\textup{aff}}(A,z)$ given by

[TABLE]

for all $1\leq i\leq n$, $1\leq j\leq n-1$, $\mathbf{a}\in A^{\otimes n}$.

Lemma 2.9**.**

Suppose that $\tau\colon A\to A^{\mathrm{op}}$ is an isomorphism of symmetric algebras (i.e. an algebra isomorphism preserving the trace map). Then the map

[TABLE]

for all $1\leq i\leq n$, $1\leq j\leq n-1$, $\mathbf{a}\in A^{\otimes n}$, is an isomorphism of algebras.

Proof.

It is straightforward to verify that $\widehat{\tau}$ preserves the defining relations of $H_{n}^{\textup{aff}}(A,z)$, once it is noted that $\widehat{\tau}(t_{i,i+1})=t_{i,i+1}$ (see the proof of [Sav20, Lem. 3.9]). So $\widehat{\tau}$ is indeed a homomorphism of algebras. That $\widehat{\tau}$ is an isomorphism follows from the fact that it has inverse $\widehat{\tau^{-1}}$. ∎

Recall that the center of $A$ is

[TABLE]

Lemma 2.10**.**

Let $a\in Z(A)$ be invertible and even. Then there exists a unique algebra automorphism $\zeta_{a}\colon H_{n}^{\textup{aff}}(A,z)\to H_{n}^{\textup{aff}}(A,z)$ given by

[TABLE]

for $\mathbf{a}\in A^{\otimes n}$, $1\leq i\leq n-1$, $1\leq j\leq n$.

Proof.

It is straightforward to verify that $\zeta_{a}$ preserves the defining relations of $H_{n}^{\textup{aff}}(A,z)$.

Then, since $\zeta_{a}$ is invertible with inverse $\zeta_{a^{-1}}$, it is an automorphism. ∎

3. Structure theory

In this section we examine the structure theory of quantum affine wreath algebras. In particular, we describe a basis, the center, Jucys–Murphy elements, and a Mackey Theorem.

3.1. Demazure Operators

Let

[TABLE]

We will use the notation $f,g$ to denote elements of $P_{n}(A)$ and the notation $p,q$ to denote elements of $P_{n}$. By abuse of notation, for $f\in P_{n}(A)$ we will also denote by $f$ its image under the natural homomorphism $P_{n}(A)\to H_{n}^{\textup{aff}}(A,z)$. In fact, it will follow from Theorem 3.10 that this homomorphism is injective, allowing us to view $P_{n}(A)$ as a subalgebra of $H_{n}^{\textup{aff}}(A,z)$.

We consider two different actions of $S_{n}$ on $P_{n}(A)$. For $w\in S_{n}$ and $f\in P_{n}(A)$, we let $w(f)$ denote the action by permuting the $X_{i}$ and superpermuting the factors of $A^{\otimes n}$, i.e. the diagonal action on $A^{\otimes n}\otimes P_{n}$. On the other hand, we let ${}^{w}f$ denote the action given by

[TABLE]

So this action permutes the $X_{i}$, but is $A^{\otimes n}$-linear. Of course $w(p)={}^{w}p$ for $p\in P_{n}$.

For $1\leq i\leq n-1$, we have the Demazure operators

[TABLE]

It is straightforward to verify that

[TABLE]

In particular,

[TABLE]

Lemma 3.1**.**

For $f\in P_{n}(A)$ and $1\leq i,j\leq n-1$, $|i-j|>1$, we have

[TABLE]

Proof.

The relations 3.5 follow from straightforward computations.

To see the first equation in 3.6, for $\mathbf{a}\in A^{\otimes n}$, $p\in P_{n}$, we compute

[TABLE]

The second equation in 3.6 is straightforward. To see 3.7, we compute

[TABLE]

For 3.8, we compute

[TABLE]

Finally, to see 3.9, we compute

[TABLE]

A similar computation for $\Delta_{i+1}\Delta_{i}\Delta_{i+1}(f)$ yields the same final expression.

∎

Remark 3.2**.**

The relations 3.7, 3.8, and 3.9 imply that the $\Delta_{i}$ define an action of the [math]-Hecke algebra on $P_{n}(A)$. Demazure operators first appeared in [Dem74]. Over the ring of integers, 3.7, 3.8, and 3.9 are proved in [Dem74, Th. 2(a)] and [Dem74, (18)].

Lemma 3.3**.**

For all $f\in P_{n}(A)$ and $1\leq i\leq n-1$, we have

[TABLE]

Proof.

It is straightforward to verify by direct computation that 3.10 holds for $f=X_{j}^{\pm 1}$, $1\leq j\leq n$. For example,

[TABLE]

Then, supposing the result holds for $p,q\in P_{n}$, we have

[TABLE]

Since both sides of 3.10 are $\Bbbk$-linear in $f$ and the $X_{j}^{\pm 1}$ generate $P_{n}$ as a $\Bbbk$-algebra, the result holds for all $p\in P_{n}$. Now, for $\mathbf{a}\in A^{\otimes n}$ and $p\in P_{n}$, we have

[TABLE]

This completes the proof. ∎

Lemma 3.4**.**

Suppose $f\in P_{n}(A)$ and $w\in S_{n}$. In $H_{n}^{\textup{aff}}(A,z)$, we have

[TABLE]

for some $f_{u},f_{u}^{\prime}\in P_{n}(A)$. Here $<$ denotes the strong Bruhat order on $S_{n}$.

Proof.

This follows from 3.10 by induction on the length of $w$. ∎

Lemma 3.5**.**

For $k,\ell\in\mathbb{Z}$, $\ell>0$, we have

[TABLE]

Proof.

We have

[TABLE]

The second equation in 3.12 then follows from the second equation in 3.5. ∎

Corollary 3.6**.**

For $1\leq i\leq n-1$, we have $\Delta_{i}(\Bbbk[X_{1},\dotsc,X_{n}])\subseteq\Bbbk[X_{1},\dotsc,X_{n}]$.

Proof.

This follows from 3.12 and the fact that $\Delta_{i}$ is linear in the $X_{j}$, $j\neq i,i+1$, by 3.4. ∎

3.2. Basis Theorem

Our next goal is to give explicit bases for the quantum affine wreath algebra $H_{n}^{\textup{aff}}(A,z)$. We do this by constructing a natural faithful representation.

Lemma 3.7**.**

For $1\leq i\leq n-1$ and $\ell>0$,

[TABLE]

Proof.

For $1\leq i\leq n-1$ and $\ell>0$, we have

[TABLE]

Proposition 3.8**.**

Let $\mathcal{H}$ be the free $\Bbbk$-module with basis $\{T_{w}:w\in S_{n}\}$, and $V=P_{n}(A)\otimes\mathcal{H}$ a tensor product of $\Bbbk$-modules. Then $V$ is an $H_{n}^{\textup{aff}}(A,z)$-module, with the action given by

[TABLE]

for $f,g\in P_{n}(A)$, $w\in S_{n}$. Here $\ell$ is the length function on $S_{n}$.

Proof.

We need to check that the action satisfies the defining relations of $H_{n}^{\textup{aff}}(A,z)$. Throughout this proof, $\mathbf{a}\in A^{\otimes n}$, $f,g\in P_{n}(A)$, $w\in S_{n}$, $1\leq i\leq n-1$, and $1\leq j\leq n$. The relation 2.10 is clearly satisfied.

Relation 2.5: We have

[TABLE]

Relation 2.8: If $j\neq i,i+1$, then

[TABLE]

Relation 2.9: First suppose that $\ell(s_{i}w)>\ell(w)$. Then

[TABLE]

On the other hand, if $\ell(s_{i}w)<\ell(w)$, then

[TABLE]

Relation 2.4: First suppose $\ell(s_{i}w)>\ell(w)$, so that $\ell(s_{i}(s_{i}w))=\ell(w)<\ell(s_{i}w)$. Then, using the fact that $s_{i}(t_{i,i+1})=t_{i,i+1}$, we have

[TABLE]

The case $\ell(s_{i}w)<\ell(w)$ is similar.

Relation 2.2: Let $|i-j|>1$, so that $s_{i}s_{j}=s_{j}s_{i}$. In order to handle several cases simultaneously, we use $\varepsilon$ and $\tau$ to denote elements of $\{0,1\}$ here. Using 3.6 and 3.4, we have

[TABLE]

Relation 2.3: Verifying 2.3 is the most involved, and it occupies the remainder of the proof. We first show that for $1\leq i\leq n-2$ and $1\leq j\leq n$ we have

[TABLE]

as operators on $V$. Clearly 3.14 holds for $j\neq i,i+1,i+2$ because 2.8 holds for the operators $T_{i}$ and $X_{j}$ on $V$, which we have already checked. Notice also that 3.10 holds in $\operatorname{End}(V)$ since the proof of that relation depends only on 2.9, 2.8, and 2.4, which we have already verified.

For $j=i$, as operators in $\operatorname{End}(V)$ we have

[TABLE]

and

[TABLE]

So 3.14 holds for $j=i$. The cases $j=i+1,i+2$ are similar.

Notice that 3.14 also implies

[TABLE]

Then 3.14, 3.15, and 2.5 imply that, as operators on $V$,

[TABLE]

Thus, for all $f\in P_{n}(A)$ and $w\in S_{n}$, we have

[TABLE]

Hence, to prove 2.3, it suffices to prove that

[TABLE]

For the remainder of the proof, to simplify the notation, we write $T_{w}$ for $1\otimes T_{w}$ and we omit $\cdot$ from the notation for the action. We also adopt the convention that operators are applied in order from right to left. For example, $T_{i}T_{i+1}T_{i}T_{w}=T_{i}(T_{i+1}(T_{i}T_{w}))$.

It follows immediately from the definition of the action that we have

[TABLE]

We split the proof of 3.16 into the following cases:

1. (a)

$\ell(s_{i}s_{i+1}s_{i}w)=\ell(w)+3$, 2. (b)

$\ell(s_{i}s_{i+1}s_{i}w)=\ell(w)+1$, 3. (c)

$\ell(s_{i}s_{i+1}s_{i}w)=\ell(w)-1$, 4. (d)

$\ell(s_{i}s_{i+1}s_{i}w)=\ell(w)-3$.

In case (a), we have

[TABLE]

Thus

[TABLE]

In case (b), we have, without loss of generality,

[TABLE]

(The other possibility is obtained by interchanging $i$ and $i+1$.) Then we have a reduced word $w=s_{i}v$ and so

[TABLE]

and

[TABLE]

So 3.16 holds.

In case (c), we have, without loss of generality,

[TABLE]

(The other possibility is obtained by interchanging $i$ and $i+1$.) Then we have a reduced word $w=s_{i}s_{i+1}v$ and so

[TABLE]

and

[TABLE]

So 3.16 holds.

The case (d) is similar and so will be omitted.

∎

Remark 3.9**.**

Notice that if $p\in\Bbbk[X_{1},\dotsc,X_{n}]$, then $X_{i+1}^{-1}\Delta_{i}(X_{i+1}p)\in\Bbbk[X_{1},\dotsc,X_{n}]$ by 3.12. Then, also using 3.6, it is easy to see that if we take $V_{+}\subsetneq V$ to be the space obtained by replacing $P_{n}(A)$ with $\Bbbk[X_{1},\dotsc,X_{n}]\otimes A$ in 3.8, then $V_{+}$ is invariant under the action of $H^{\textup{aff}}_{n,+}(A,z)$.

Theorem 3.10** (Basis Theorem for $H_{n}^{\textup{aff}}(A,z)$).**

The map

[TABLE]

is an isomorphism of $H_{n}^{\textup{aff}}(A,z)$-modules.

Proof.

Let $D$ be a basis of $P_{n}(A)$, and let

[TABLE]

Then $\mathcal{B}_{1}$ is a basis of $V$. It follows from Lemma 3.3 that $\mathcal{B}_{2}$ spans $H_{n}^{\textup{aff}}(A,z)$. Furthermore, we have that $(fT_{w})\cdot(1\otimes T_{1})=f\otimes T_{w}$, and so the elements of $\mathcal{B}_{2}$ are linearly independent, hence a basis. Since $V$ is a cyclic module generated by $1\otimes T_{e}$, there is an $H_{n}^{\textup{aff}}(A,z)$-module homomorphism $H_{n}^{\textup{aff}}(A,z)\to V$, determined by $1\mapsto 1\otimes T_{e}$. This map sends $fT_{w}\in\mathcal{B}_{2}$ to $f\otimes T_{w}\in\mathcal{B}_{2}$, hence it is an isomorphism because it gives a bijection of $\Bbbk$-bases. ∎

For $\lambda=(\lambda_{1},\dotsc,\lambda_{n})\in\mathbb{Z}^{n}$, we let $X^{\lambda}=X_{1}^{\lambda_{1}}\dotsm X_{n}^{\lambda_{n}}$. Recall that $B$ is a $\Bbbk$-basis for $A$.

Corollary 3.11**.**

The sets

[TABLE]

are $\Bbbk$-bases for $H_{n}^{\textup{aff}}(A,z)$.

Proof.

It follows immediately from Theorem 3.10 that the first set is a basis. The fact that the second set is also a basis follows from 3.11 by induction on the length of $w$. ∎

Corollary 3.12**.**

The sets

[TABLE]

are $\Bbbk$-bases for $H_{n,+}^{\textup{aff}}(A,z)$.

Proof.

This uses the same reasoning as 3.11, due to 3.9. ∎

Remark 3.13**.**

For the case of the affine Hecke algebras (see Example 2.6), 3.11 recovers a result of Lusztig [Lus89, Prop. 3.7]. For affine Yokonuma–Hecke algebras (see Example 2.7), it was proved in [CPd16, Th. 4.4].

3.3. Description of the center

We now compute the center of quantum affine wreath algebras. By 2.10, we have that $P_{n}(Z(A))=Z(A)^{\otimes n}\otimes P_{n}$ is a subalgebra of $P_{n}(A)$.

Lemma 3.14**.**

The centralizer of $P_{n}$ in $H_{n}^{\textup{aff}}(A,z)$ is equal to $P_{n}(A)$.

Proof.

By 2.10, it is clear that elements of $P_{n}(A)$ commute with elements of $P_{n}$. Now let $\alpha=\sum_{w\in S_{n}}f_{w}T_{w}\in H_{n}^{\textup{aff}}(A,z)$, where $f_{w}\in P_{n}(A)$ for all $w\in S_{n}$. Let $v\in S_{n}$ be a maximal element in the strong Bruhat order such that $f_{v}\neq 0$. Suppose $v\neq 1$, and let $1\leq i\leq n$ such that $v(i)\neq i$. Then, by 3.11 we have

[TABLE]

for some $g_{u}\in P_{n}(A)$. Thus, by Theorem 3.10, $\alpha$ does not centralize $X_{i}$; hence the result. ∎

Lemma 3.15**.**

The centralizer of $P_{n}(A)$ in $H_{n}^{\textup{aff}}(A,z)$ is equal to $P_{n}(Z(A))$.

Proof.

The centralizer of $P_{n}(A)$ inside $H_{n}^{\textup{aff}}(A,z)$ is contained in the centralizer of $P_{n}$, which by Lemma 3.14 is equal to $P_{n}(A)$. Hence the centralizer of $P_{n}(A)$ is equal to the center $Z(P_{n}(A))$. Using 2.10, we have

[TABLE]

where we use the fact that $Z(A^{\otimes n})=Z(A)^{\otimes n}$ since $A$ is free over $\Bbbk$. ∎

For any subset $Y\subseteq P_{n}(A)$, we define

[TABLE]

Theorem 3.16**.**

We have Z\big{(}H_{n}^{\textup{aff}}(A,z)\big{)}=P_{n}(Z(A))^{S_{n}}.

Proof.

Suppose $f\in Z(H_{n}^{\textup{aff}}(A,z))\subseteq P_{n}(Z(A))$. For $1\leq i\leq n-1$, 3.11 implies that

[TABLE]

Then

[TABLE]

By 3.11, we have $g=0$; hence $f=s_{i}(f)$. Since this is true for all $1\leq i\leq n-1$, it follows that $f\in P_{n}(Z(A))^{S_{n}}$.

Now suppose $f\in P_{n}(Z(A))^{S_{n}}$. For each $1\leq i\leq n-1$ and $\mathbf{a}\in Z(A)^{\otimes n}$, we have

[TABLE]

It follows that $t_{i,i+1}{}^{s_{i}}f=t_{i,i+1}s_{i}(f)$, and so

[TABLE]

Thus, by 3.10, we have $T_{i}f=fT_{i}$ for all $1\leq i\leq n-1$. Since $f$ clearly commutes with all elements of $P_{n}(A)$, we have f\in Z\big{(}H_{n}^{\textup{aff}}(A,z)\big{)}. ∎

Remark 3.17**.**

For affine Hecke algebras (see Example 2.6), Theorem 3.16 recovers a well-known description of the center (see [Lus89, Prop. 3.11]). For affine Yokonuma–Hecke algebras (see Example 2.7), it recovers [CW, Th. 2.7].

Proposition 3.18**.**

Suppose $A^{\prime}$ is a maximal commutative subalgebra of $A$. Then $P_{n}(A^{\prime})$ is a maximal commutative subalgebra of $H_{n}^{\textup{aff}}(A,z)$.

Proof.

Suppose $\alpha\in H_{n}^{\textup{aff}}(A,z)$ commutes with all elements of $P_{n}(A^{\prime})$. By Lemma 3.14, we have $\alpha\in P_{n}(A)$. Thus $\alpha=\sum_{\lambda\in\mathbb{Z}^{n}}\mathbf{a}_{\lambda}X^{\lambda}$, for some $\mathbf{a}_{\lambda}\in A^{\otimes n}$. Then, for all $\mathbf{b}\in(A^{\prime})^{\otimes n}$, we have

[TABLE]

Thus, by 3.11, $\mathbf{b}\mathbf{a}_{\lambda}=\mathbf{a}_{\lambda}\mathbf{b}$ for all $\lambda$. Since $A^{\prime}$ is a maximal commutative subalgebra of $A$, this implies that $\mathbf{a}_{\lambda}\in(A^{\prime})^{\otimes n}$ for all $\lambda$. Hence $\alpha\in P_{n}(A^{\prime})$. ∎

3.4. Jucys–Murphy elements

Define the Jucys–Murphy elements in $H_{n}(A,z)$ by

[TABLE]

These elements generalize the well-known Jucys–Murphy elements in the Iwahori–Hecke algebra, as well as the Jucys–Murphy elements of the Yokonuma–Hecke algebra introduced in [CPd14, (2.14)].

Proposition 3.19**.**

There is a surjective algebra homomorphism $H_{n}^{\textup{aff}}(A,z)\to H_{n}(A,z)$ defined by

[TABLE]

Proof.

We need to check that this maps preserves the relations 2.8, 2.9, and 2.10, in addition to the facts that $X_{i}X_{j}=X_{j}X_{i}$ and that $X_{i}$ is invertible for $1\leq i,j\leq n$. Clearly $J_{i}$ is invertible for all $1\leq i\leq n$ because $T_{k}$ is invertible for all $1\leq k\leq n$. Also, 2.9 follows from the definition of $J_{i}$. Relation 2.10 is satisfied because of 2.5, while the fact that $J_{i}J_{j}=J_{j}J_{i}$ for all $1\leq i,j\leq n$, in addition to relation 2.8, follows from repeated use of 2.2 and 2.3. ∎

3.5. Mackey Theorem

For a composition $\mu=(\mu_{1},\dotsc,\mu_{r})$ of $n$, let

[TABLE]

denote the corresponding Young subgroup. We then define the parabolic subalgebra $H_{\mu}(A,z)\subseteq H_{n}(A,z)$ to be the subalgebra generated by $A^{\otimes n}$ and $\{T_{w}:w\in S_{\mu}\}$. We also define $H_{\mu}^{\textup{aff}}(A,z)\subseteq H_{n}^{\textup{aff}}(A,z)$ to be the subalgebra generated by $H_{\mu}(A,z)$ and $P_{n}$. So we have an isomorphism of algebras

[TABLE]

and a parity-preserving isomorphism of $\Bbbk$-modules

[TABLE]

Let $D_{\mu,\nu}$ denote the set of minimal length $(S_{\mu},S_{\nu})$-double coset representatives in $S_{n}$. By [DJ86, Lem. 1.6(ii)], for $\pi\in D_{\mu,\nu}$, $S_{\mu}\cap\pi S_{\nu}\pi^{-1}$ and $\pi^{-1}S_{\mu}\pi\cap S_{\nu}$ are Young subgroups of $S_{n}$; hence we can define compositions $\mu\cap\pi\nu$ and $\pi^{-1}\mu\cap\nu$ by

[TABLE]

Furthermore, the map $w\mapsto\pi^{-1}w\pi$ restricts to a length preserving isomorphism

[TABLE]

which, due to the length-preserving property, induces an isomorphism of algebras

[TABLE]

It is easy to verify that, for $\pi\in D_{\mu,\nu}$ and $s_{i}\in S_{\mu\cap\pi\nu}$, we have $\pi^{-1}(i+1)=\pi^{-1}(i)+1$, and hence $\pi^{-1}s_{i}\pi=s_{\pi^{-1}(i)}$.

Thus, for each $\pi\in D_{\mu,\nu}$, we have an algebra isomorphism

[TABLE]

If $N$ is a left $H_{\pi^{-1}\mu\cap\nu}^{\textup{aff}}(A,z)$-module, we denote by ${}^{\pi}N$ the left $H_{\mu\cap\pi\nu}^{\textup{aff}}(A,z)$-module with action given by

[TABLE]

The inclusion $H_{\mu}^{\textup{aff}}(A,z)\subseteq H_{n}^{\textup{aff}}(A,z)$ gives induction and restriction functors

[TABLE]

Theorem 3.20** (Mackey Theorem for $H_{n}^{\textup{aff}}(A,z)$).**

Suppose that $M$ is an $H_{n}^{\textup{aff}}(A,z)$-module. Then $\operatorname{Res}^{n}_{\mu}\operatorname{Ind}^{n}_{\nu}M$ admits a filtration with subquotients evenly isomorphic to

[TABLE]

one for each $\pi\in D_{\mu,\nu}$. Furthermore, the subquotients can be taken in any order refining the strong Bruhat order on $D_{\mu,\nu}$. In particular, $\operatorname{Ind}^{\mu}_{\mu\cap\nu}\operatorname{Res}^{\nu}_{\mu\cap\nu}M$ appears as a submodule.

Proof.

The proof is essentially the same as the proofs of [Kle05, Thm 3.5.2] and [Kle05, Thm 14.5.2]; hence it will be omitted. ∎

4. Cyclotomic quotients

In this final section we define cyclotomic quotients of the quantum affine wreath algebras and prove some of their key properties. These quotients are natural analogues of cyclotomic quotients of affine Hecke algebras (see Example 2.6).

4.1. Definitions

Identifying $A$ with $A\otimes 1^{\otimes n-1}$, we can naturally view $P_{1}(A)=A[X_{1}]$ as a subalgebra of $P_{n}(A)$. Let

[TABLE]

be a monic polynomial of degree $d$ in $X_{1}$ with coefficients in $Z(A)_{0}$, the even part of the center of $A$. We write

[TABLE]

with $a_{(i)}\in Z(A)_{0}$. We assume that $a_{(0)}$ is invertible.

We define the corresponding quantum cyclotomic wreath algebra to be

[TABLE]

where $\langle f\rangle$ denotes the two-sided ideal in $H_{n}^{\textup{aff}}(A,z)$ generated by $f$. We call $d$ the level of $H_{n}^{f}(A,z)$. Since $f\in H_{n,+}^{\textup{aff}}(A,z)\subseteq H_{n}^{\textup{aff}}(A,z)$, we can also define

[TABLE]

where $\langle f\rangle_{+}$ denotes the two-sided ideal in $H_{n,+}^{\textup{aff}}(A,z)$ generated by $f$.

Let $f_{1}:=f$ and, for $2\leq i\leq n$, define

[TABLE]

It follows immediately that

[TABLE]

Lemma 4.1**.**

For $1\leq i\leq n-1$, we have

[TABLE]

where $\Bbbk[X_{1},\dotsc,X_{i-1}]_{\leq d-e}$ denotes the space of polynomials of degree less than or equal to $d-e$.

Proof.

We prove this by induction. For $i=1$, the result is immediate. Now assuming the result true for all $1\leq j\leq i$, we have

[TABLE]

where, for the final inclusion, we used 3.13. ∎

Consider the algebra homomorphism given by the composition

[TABLE]

where the first map is the natural inclusion and the second is the projection.

Lemma 4.2**.**

The map $\eta$ is surjective.

Proof.

Notice that

[TABLE]

and so

[TABLE]

It then follows by induction that $X_{i+1}^{-1}=T_{i}^{-1}X^{-1}_{i}T_{i}^{-1}\in\eta\left(H^{\textup{aff}}_{n,+}(A,z)\right)$ for all $1\leq i\leq n-1$, which gives the result. ∎

Lemma 4.3**.**

We have

[TABLE]

Proof.

We have

[TABLE]

where the sixth equality follows from the fact that the $T_{j}$ are invertible. The final equality in the statement of the lemma then follows from 4.3. ∎

Lemma 4.4**.**

For all $1\leq i\leq n$, we have

[TABLE]

Proof.

By Lemma 4.3, any element of the intersection is of the form

[TABLE]

for some $\mathbf{a}_{j,w}\in A^{\otimes n}$. It follows from 3.11 and 4.1 that $\mathbf{a}_{j,w}=0$ whenever $j\neq i$. Then, by 3.13, the constant term (i.e. the term of degree zero in the $X_{j}$) of $\sum_{w}\mathbf{a}_{i,w}f_{i}T_{w}$, which must equal zero, is

[TABLE]

Since the $T_{j}$ are invertible, as is $a_{(0)}$, it follows from 3.11 that $\mathbf{a}_{i,w}=0$ for all $w\in S_{n}$. ∎

Proposition 4.5**.**

The map $\eta$ induces an isomorphism

[TABLE]

Proof.

We need to show that $\langle f\rangle_{+}=\eta^{-1}(\langle f\rangle)$, which is to say that

[TABLE]

Clearly $\langle f\rangle_{+}\subseteq\langle f\rangle\cap H_{n,+}^{\textup{aff}}(A,z)$; so we need to show the other inclusion.

Define a partial order on $\mathbb{Z}^{n}$ by $\lambda\leq\mu$ if $\lambda_{i}\leq\mu_{i}$ for all $1\leq i\leq n$. Using Lemmas 3.4 and 2.5, any element of the form

[TABLE]

can be written in the form $\sum_{\lambda,w,v}\mathbf{c}_{\lambda,w,v}X^{\lambda}T_{w}fT_{v}$. Thus, it suffices to show that if

[TABLE]

then $\lambda\geq 0$ whenever $\mathbf{c}_{\lambda,w,v}\neq 0$ for some $w,v\in S_{n}$.

Take a minimal element $\lambda=(\lambda_{1},\dotsc,\lambda_{n})$ appearing in 4.5 with $\mathbf{c}_{\lambda,w,v}\neq 0$. Suppose, towards a contradiction, that $\lambda_{i}<0$ for some $1\leq i\leq n$. Since $f\in A[X_{1}]$, by Lemmas 3.3 and 3.6 we have

[TABLE]

for all $w,v$. Thus, by 3.11 and the minimality of $\lambda$, we must have

[TABLE]

Since $\lambda_{i}<0$, 3.11 implies that $\sum_{w,v}\mathbf{c}_{\lambda,w,v}T_{w}fT_{v}\in X_{i}H_{n,+}^{\textup{aff}}(A,z)$. But this is impossible by Lemma 4.4. ∎

4.2. Basis Theorem

We now prove a basis theorem for $H^{f}_{n,+}(A,z)$, which also gives a basis theorem for $H^{f}_{n}(A,z)$ in light of 4.5. We follow the methods of [Kle05, §7.5] and [Sav20, §6.3].

For $I=\{i_{1}<\dotsb<i_{k}\}\subseteq\{1,\dotsc,n\}$, let

[TABLE]

We also define

[TABLE]

Lemma 4.6**.**

We have that $H_{n,+}^{\textup{aff}}(A,z)$ is a free right $H_{n}(A,z)$-module with basis

[TABLE]

Proof.

Consider the lexicographic ordering $\prec$ on $\mathbb{N}^{n}$. Define a function

[TABLE]

Using induction on $n$ and Lemma 4.1, we see that, for all $(\lambda,I)\in\Omega_{n}$,

[TABLE]

Now, $\gamma\colon\Omega_{n}\to\mathbb{N}^{n}$ is a bijection and, by 3.12, $\{X^{\lambda}:\lambda\in\mathbb{N}^{n}\}$ is a basis for $H_{n,+}^{\textup{aff}}(A,z)$ as a right $H_{n}(A,z)$-module. Thus the lemma follows from 4.6. ∎

Lemma 4.7**.**

We have that $f_{n}$ commutes with all elements of $H_{n-1}(A,z)$.

Proof.

It follows from the definition 4.2 of $f_{n}$ and from the relations 4.3, 2.3, and 2.8 that $f_{n}$ commutes with $T_{i}$, $1\leq i\leq n-2$, and $\mathbf{a}\in A^{\otimes(n-1)}$.

∎

Lemma 4.8**.**

We have $\langle f\rangle_{+}=\sum_{i=1}^{n}\Bbbk[X_{1},\dotsc,X_{n}]f_{i}H_{n}(A,z)$.

Proof.

We have

[TABLE]

Lemma 4.9**.**

For $d>0$, we have $\langle f\rangle_{+}=\sum_{(\lambda,I)\in\Omega_{n}^{+}}X^{\lambda}f_{I}H_{n}(A,z)$.

Proof.

We prove this by induction on $n$. When $n=1$, the statement is obvious. Now suppose that $n>1$, and define $\langle f\rangle^{\prime}_{+}:=H_{n-1,+}^{\textup{aff}}(A,z)fH_{n-1,+}^{\textup{aff}}(A,z)$. By the induction hypothesis we have

[TABLE]

Let $J=\sum_{(\lambda,I)\in\Omega_{n}^{+}}X^{\lambda}f_{I}H_{n}(A,z)$. Clearly $J\subseteq\langle f\rangle_{+}$, and so we need to show that $\langle f\rangle_{+}\subseteq J$. By Lemma 4.8, it is enough to show that $X^{\lambda}f_{i}H_{n}(A,z)\subseteq J$ for all $\lambda\in\mathbb{N}^{n}$ and $1\leq i\leq n$. Consider first the case $i=n$ and write $X^{\lambda}=X_{n}^{\lambda_{n}}X^{\mu}$, where $\mu=(\lambda_{1},\dotsc,\lambda_{n-1})\in\mathbb{N}^{n-1}$. Expanding $X^{\mu}$ in terms of the basis of $H_{n-1,+}^{\textup{aff}}(A,z)$ of Lemma 4.6, we see that

[TABLE]

where the second inclusion follows from Lemma 4.7.

Now consider $X^{\lambda}f_{i}H_{n}(A,z)$, with $1\leq i<n$. As above, we write $X^{\lambda}=X_{n}^{\lambda_{n}}X^{\mu}$, where $\mu\in\mathbb{N}^{n-1}$. By the induction hypothesis, we have

[TABLE]

Now we show by induction on $\lambda_{n}$ that $X_{n}^{\lambda_{n}}X^{\lambda^{\prime}}f_{I^{\prime}}H_{n}(A,z)\in J$ for all $(\lambda^{\prime},I^{\prime})\in\Omega_{n-1}^{+}$. This follows immediately from the definition of $\Omega_{n-1}^{+}$ when $\lambda_{n}<d$. If $\lambda_{n}\geq d$, by Lemma 4.1 we have

[TABLE]

By the definition of $J$, we have $X_{n}^{\lambda_{n}-d}X^{\lambda^{\prime}}f_{I^{\prime}}f_{n}H_{n}(A,z)\in J$. Now, by 4.7, for $0\leq e<d$, we have

[TABLE]

Since $0\leq\lambda_{n}-d+e<\lambda_{n}$, each term in the above sum is contained in $J$ by the induction hypothesis, which concludes the proof. ∎

Theorem 4.10** (Basis theorem for cyclotomic quotients).**

The canonical images of the elements

[TABLE]

form a basis of $H_{n,+}^{f}(A,z)$ and of $H_{n}^{f}(A,z)$.

Proof.

By Lemmas 4.6 and 4.9, the elements $\{X^{\lambda}f_{I}:(\lambda,I)\in\Omega_{n}^{+}\}$ form a basis for $\langle f\rangle_{+}$ as a $H_{n}(A,z)$-right module. Thus Lemma 4.6 implies that

[TABLE]

is a basis for a complement to $\langle f\rangle_{+}$ inside $H_{n,+}^{\textup{aff}}$, viewed as a right $H_{n}(A,z)$-module. ∎

Remark 4.11**.**

In the setting of affine Hecke algebras (see Example 2.6), Theorem 4.10 recovers [AK94, Th. 3.10]. For affine Yokonuma–Hecke algebras (see Example 2.7), it was proved in [CPd16, Th. 4.4].

Corollary 4.12**.**

Every level one quantum cyclotomic wreath algebra is isomorphic to $H_{n}(A,z)$.

Proof.

If $f=X_{1}-1$, then the map $H_{n}^{\textup{aff}}(A,z)\twoheadrightarrow H^{f}_{n}(A,z)\cong H_{n}(A,z)$ is exactly the map of 3.19. In general $f=X_{1}+a$ with $a\in Z(A)$ even and invertible. So the result follows by applying the automorphism $\zeta_{-a}$ of Lemma 2.10. ∎

4.3. Cyclotomic Mackey Theorem

Theorem 4.10 implies that the subalgebra of $H_{n+1}^{f}(A,z)$ generated by $X_{1},\dotsc,X_{n}$, $A^{\otimes n}\otimes 1$ and $T_{1},\dotsc,T_{n-1}$ is isomorphic to $H_{n}^{f}(A,z)$. Thus we can define induction and restriction functors

[TABLE]

Let $\Pi\colon H_{n}^{f}(A,z)\textup{-mod}\to H_{n}^{f}(A,z)\textup{-mod}$ denote the functor that reverses the parity of the elements of a module.

Proposition 4.13**.**

Recall that $d=\deg f$.

1. (a)

We have that $H^{f}_{n+1}(A,z)$ is a free right $H^{f}_{n}(A,z)$-module with basis

[TABLE] 2. (b)

We have a decomposition of $(H_{n}^{f}(A,z),H_{n}^{f}(A,z))$-bimodules

[TABLE] 3. (c)

For $0\leq r\leq d$ and homogeneous $a\in A$, we have parity-preserving isomorphisms of $(H_{n}^{f}(A,z),H_{n}^{f}(A,z))$-bimodules

[TABLE]

Proof.

The proof is almost identical to the proof of [Kle05, Lemma 7.6.1] and so will be omitted. ∎

Theorem 4.14** (Cyclotomic Mackey Theorem).**

For all $n\in\mathbb{N}_{+}$, we have a natural isomorphism of functors

[TABLE]

Proof.

This follows from 4.13. ∎

Remark 4.15**.**

4.13 is the key ingredient in showing that the quantum Frobenius Heisenberg categories of [BSW] act on categories of modules for quantum cyclotomic wreath algebras. It corresponds to the inversion relation in the quantum Frobenius Heisenberg categories.

4.4. Symmetric algebra structure

By Theorem 4.10, we can define a $\Bbbk$-linear map

[TABLE]

where $\mathrm{tr}(\mathbf{a})=\mathrm{tr}^{\otimes n}(\mathbf{a})$ is the natural trace map on the tensor product algebra $A^{\otimes n}$ (here, on the right-hand side, $\mathrm{tr}$ is the trace map on $A$).

Theorem 4.16**.**

The cyclotomic quotient $H_{n}^{f}(A,z)$ is a symmetric algebra with trace map $\mathrm{tr}_{f}^{n}$.

Proof.

Consider the total order on $\mathbb{N}^{n}$ given by $\lambda<\mu$ if and only if

[TABLE]

For the remainder of this proof,

• •

$\lambda$ and $\mu$ will denote elements of $\mathbb{N}^{n}$ such that $\lambda_{i},\mu_{i}<d$ for all $i$,

• •

$\mathbf{a}$ and $\mathbf{b}$ will denote elements of $B^{\otimes n}$, and

• •

$u,v,w$ will denote elements of $S_{n}$.

We must verify that the basis given in Theorem 4.10 has a left dual basis with respect to $\mathrm{tr}_{f}^{n}$. By 3.11, we have

[TABLE]

for some $f_{u}\in P_{n}(A)$.

By Theorem 3.10, the equation 3.17 holds in $H_{n}^{\textup{aff}}(A,z)$. It follows that

[TABLE]

The second equation above also implies that

[TABLE]

since $u<v^{-1}\implies u^{-1}<v$. Thus, it follows from 4.10 and 4.1 that $\mathrm{tr}_{f}^{n}(X^{-\lambda}\mathbf{a}^{\vee}T_{w^{-1}}X^{\lambda}\mathbf{a}T_{w})=1$ and that

[TABLE]

whenever

• •

$w<v$, or

• •

$w=v$ and $\lambda<\mu$, or

• •

$w=v$, $\lambda=\mu$, and $\mathbf{a}\neq\mathbf{b}$.

Thus we can find a left dual basis to the basis given in Theorem 4.10 by inverting a unitriangular matrix.

It remains to prove that the trace map $\mathrm{tr}_{f}^{n}$ is symmetric. Let $\psi$ denote the Nakayama automorphism corresponding to $\mathrm{tr}_{f}^{n}$ (see 2.8). So we want to show that $\psi$ is the identity automorphism. It follows from 2.5 and 2.10 that

[TABLE]

So $\psi(\mathbf{b})=\mathbf{b}$.

If $\lambda\neq 0$, we have (noting that $\Delta_{i}$ preserves polynomial degree)

[TABLE]

We also have

[TABLE]

Similarly,

[TABLE]

Thus $\psi(T_{i})=T_{i}$.

Now, if $\psi(X_{i})=X_{i}$ for some $1\leq i\leq n-1$, then

[TABLE]

Therefore, it remains to show that $\psi(X_{1})=X_{1}$. That is, we need to show

[TABLE]

It follows from 2.8 and 2.10 that 4.11 holds when $w(1)=1$.

Now suppose $w=s_{1}v$ for some $v\in S_{n}$ with $v(1)=1$. Then

[TABLE]

4.5. Frobenius extension structure

Let

[TABLE]

By 4.13(b), we have a decomposition of $(H_{n}^{f}(A,z),H_{n}^{f}(A,z))$-bimodules

[TABLE]

Define the partial trace map

[TABLE]

to be the homomorphism of $(H_{n}^{f}(A,z),H_{n}^{f}(A,z))$-bimodules given by the projection onto the first summand in 4.15 followed by the map

[TABLE]

It follows that

[TABLE]

Proposition 4.18**.**

The quantum cyclotomic wreath algebra $H_{n+1}^{f}(A,z)$ is a Frobenius extension of $H_{n}^{f}(A,z)$ with trace map $\mathrm{tr}_{n+1}^{f}$.

Proof.

Since $H_{n}^{f}(A,z)\subseteq H_{n+1}^{f}(A,z)$ are both symmetric algebras, it follows from [PS16, Cor. 7.4] that $H_{n+1}^{f}(A,z)$ is a Frobenius extension of $H_{n}^{f}(A,z)$ with trace map

[TABLE]

where $Y$ is a basis of $H_{n}^{f}(A,z)$. (Note that $\beta^{\vee}$ denotes the right dual of $\beta$ in [PS16], whereas it denotes the left dual in the current paper.) Since

[TABLE]

the result follows. ∎

It follows from 4.18 that the functors ${}^{f}\operatorname{Ind}_{n}^{n+1}$ and ${}^{f}\operatorname{Res}_{n}^{n+1}$ are both left and right adjoint to each other. Indeed, induction is always left adjoint to restriction. It is also right adjoint to restriction precisely when the larger algebra is a Frobenius extension of the smaller.