2.1 Wreath products

Let $\Sigma _d$ be the symmetric group on the set $[d] := \{1, 2, \ldots , d\}$ generated by $S = \{s_1, \ldots , s_{d-1}\}$ consisting of simple transpositions $s_i = (i, i+1)$ . Define

(2.1.1) $$ \begin{align} s_{a\to b} = \begin{cases} s_a s_{a-1} \cdots s_{b+1} s_b &\text{if } a> b; \\ {s_a }&{\text{if } a=b;} \\ s_a s_{a+1} \cdots s_{b-1} s_b &\text{if } a < b, \end{cases} \quad { s_{a\to b \to a} = \begin{cases} s_a s_{a-1} \cdots s_{b+1} s_b s_{b+1} \cdots s_a&\text{if } a> b; \\ {s_a }&{\text{if } a=b;} \\ s_a s_{a+1} \cdots s_{b-1} s_b s_{b-1} \cdots s_a&\text{if } a < b. \end{cases}} \end{align} $$

Recall that for a group G, the wreath product $G \wr \Sigma _d$ is the semidirect product $G^d \rtimes \Sigma _d$ whose multiplication rule is determined by

(2.1.2) $$ \begin{align} (g_1, \ldots, g_d)w = w(g_{w(1)}, \ldots, g_{w(d)}) \quad \text{for all} \quad w \in \Sigma_d, \ g_i \in G. \end{align} $$

When G is finite, say $G \subseteq \Sigma _m$ for some m, then the direct product $G^d \subseteq \Sigma _m^d \subseteq \Sigma _{md}$ . Thus, $G \wr \Sigma _d$ can be regarded as a subgroup of $\Sigma _{md}$ by identifying each $w \in \Sigma _d$ with the following permutation:

(2.1.3) $$ \begin{align} (i-1)m + j \mapsto (w(i)-1)m+j \quad \text{for all} \quad 1\leq i \leq d, 1\leq j \leq m. \end{align} $$

The complex reflection group $G(m,1,d)$ affords two nonisomorphic deformations – the Ariki–Koike algebras (or cyclotomic Hecke algebras, see Section A.4), and the Yokonuma–Hecke algebras.

Example 2.1.2 ( $\Sigma _m \wr \Sigma _2$ and Weyl groups of type D).

The wreath product $\Sigma _m \wr \Sigma _2$ is now identified with the subgroup of $\Sigma _{2m}$ generated by the Young subgroup $\Sigma _m^2 := \langle s_{(i-1)m+1}, \ldots , s_{im-1}~|~1\leq i\leq 2\rangle \subseteq \Sigma _{2m}$ , as well as the element $t_1=w_{m,m} \in \Sigma _{2m}$ (which corresponds to the generator of $\Sigma _2$ via Equation (2.1.3)), where $w_{a,b}\in \Sigma _{a+b}$ is now the element given by, in two-line notation,

(2.1.4) $$ \begin{align} w_{a,b} = \left( \begin{array}{cccccc} 1 &\cdots& a &a+1& \cdots& a+b \\ b+1& \cdots& b+a &1& \cdots &b \end{array} \right). \end{align} $$

It is well known that $w_{a,b} = s_{a\to 1} s_{a+1 \to 2} \dots s_{a+b-1 \to b}$ is a reduced expression and that

(2.1.5) $$ \begin{align} w_{a,b} s_i= \begin{cases} s_{i+a} w_{a,b} &\text{if } i<b; \\ s_{i-b} w_{a,b} &\text{if } i>b. \end{cases} \end{align} $$

Consequently, we can treat both $\Sigma _m \wr \Sigma _2$ and $\Sigma _{2m}$ as subgroups of a larger wreath product, that is, the signed permutation group on $[\pm 2m]$ , and then obtain the following tower of groups:

(2.1.6) $$ \begin{align} \Sigma_m \times \Sigma_m \subseteq \Sigma_m \wr \Sigma_2 \subseteq \Sigma_{2m} \subseteq W(\mathrm{B}_{2m}) = \Sigma_2 \wr \Sigma_{2m}. \end{align} $$

Finally, we denote by $W(\mathrm {D}_r)$ the subgroup of $W(\mathrm {B}_r)$ generated by $s_1, \ldots , s_{r-1}$ and $s_u := s_0s_1s_0$ . Indeed, $W(\mathrm {D}_r)$ is a Weyl group of type $\mathrm {D}_r$ with the following Dynkin diagram:

2.2 Affine symmetric groups

Within this section, we allow G to be an infinite monoid in a wreath product $G \wr \Sigma _d$ . Now, we consider the weight lattice $P = \sum _{i=1}^d \mathbb {Z} \epsilon _i$ for $\mathrm {GL}_d$ . The lattice P contains the root lattice $Q = \sum _{i=1}^{d-1} \mathbb {Z} \alpha _i$ , where $\alpha _i = \epsilon _{i} - \epsilon _{i+1}$ . We further let $P^+ := \sum _{i=1}^d \mathbb {N} \epsilon _i$ . Denote the corresponding translation monoids by $t^L =\{ t^\lambda ~|~ \lambda \in L\}$ for $L\in \{P, P^+, Q\}$ , with $t^\lambda t^\mu = t^{\lambda +\mu }$ for all $\lambda , \mu \in L$ .

The extended and affine symmetric groups are the semidirect products $\Sigma ^{\mathrm {aff}}_d := t^Q \rtimes \Sigma _d$ and $\Sigma ^{\mathrm {ext}}_d := t^P \rtimes \Sigma _d$ , respectively, whose multiplications are uniquely determined by $ t^\lambda s_i = s_i t^{s_i(\lambda )}. $ Next, under the identification $P \equiv \mathbb {Z}^d, \lambda = \sum \lambda _i \epsilon _i \mapsto (\lambda _1, \dots , \lambda _d)$ , we see that the simple reflection $s_i$ on P agrees with the place permutation $s_i$ on $\mathbb {Z}^d$ :

(2.2.1) $$ \begin{align} \begin{aligned} s_i(\lambda) &= \lambda - \langle\lambda, \alpha_i\rangle\alpha_i = \lambda - (\lambda_i - \lambda_{i+1})(\epsilon_i - \epsilon_{i+1}) \\ &= (\lambda_1, \dots, \lambda_{i-1},\lambda_{i+1}, \lambda_i,\lambda_{i+2}, \dots, \lambda_d). \end{aligned} \end{align} $$

Therefore, $\Sigma ^{\mathrm {ext}}_d$ can be realized as the wreath product $\mathbb {Z} \wr \Sigma _d$ whose multiplication is determined uniquely by $ (\lambda _1, \dots , \lambda _d)s_i = s_i(\lambda _1, \dots , \lambda _{i-1},\lambda _{i+1}, \lambda _i,\lambda _{i+2}, \dots , \lambda _d)$ ; while $\Sigma ^{\mathrm {aff}}_d$ is a subgroup given by

(2.2.2) $$ \begin{align} \Sigma^{\mathrm{aff}}_d = \{(\lambda_1, \dots, \lambda_d)w \in \mathbb{Z}\wr\Sigma_d ~|~ \Sigma_i \lambda_i = 0\} \subsetneq \mathbb{Z} \wr \Sigma_d. \end{align} $$

Note that we can thus regard the extended affine Hecke algebra (or degenerate affine Hecke algebra, resp.) of type A as a quantization of the wreath product $\mathbb {Z} \wr \Sigma _d = \Sigma ^{\mathrm {ext}}_d$ (or the submonoid $\mathbb {N} \wr \Sigma _d \subseteq \mathbb {Z} \wr \Sigma _d$ , resp.) as in Table 1.

Remark 2.2.1. $\Sigma ^{\mathrm {aff}}_d$ is a Coxeter group with generators $s_0, \dots , s_d$ and the following Dynkin diagram:

The extra generator $s_0$ is obtained as follows: Let $\theta := \alpha _1 + \dots + \alpha _{d-1} = \epsilon _1 - \epsilon _d$ be the highest root. Then

(2.2.3) $$ \begin{align} s_0 = t^\theta s_\theta = (1, 0, \dots, 0, -1) s_\theta = s_\theta (-1, 0, \dots, 0, 1) = s_\theta t^{-\theta}. \end{align} $$

2.3 Iwahori–Hecke algebras

Let $q\in K^\times $ . For any Coxeter system $(W,S)$ , the Hecke algebra $\mathcal {H}_q(W)$ is the associative K-algebra generated by $T_s (s\in S)$ , subject to the braid relations for $(W,S)$ as well as the quadratic relations $T_s^2 = (q-1)T_s + q$ for $s\in S$ . For the Weyl group $W(B_n)$ of type B, we will also consider its multiparameter variant $\mathcal {H}_{(Q,q)}(W(\mathrm {B}_{2m}))$ subject to the same braid relations but with quadratic relations $T_i^2 = (q-1) T_i + q$ (for $i>0$ ) and $T_0^2 = (Q-1)T_0 + Q$ , where $Q\in K^\times $ .

2.4 Augmented algebras

Following [Reference Ginzburg and KumarGK93], by an augmented algebra we mean a K-algebra B equipped with a counit $\epsilon : B\to K$ . Denote by $B^+ := \ker \epsilon \unlhd B$ the augmentation ideal of B. Assume that $B \leq A$ is a normal subalgebra (i.e., $AB^+ = B^+A$ ). Then one can form the quotient algebra (which is also augmented), [Reference Ginzburg and KumarGK93, 5.2],

(2.4.1) $$ \begin{align} A//B := A/B^+A. \end{align} $$

3.1 The definition

For an element $Z \in B\otimes B$ and for $1\leq i \leq d-1$ , set

(3.1.1) $$ \begin{align} {Z_i} := \underset{i-1 \text{ factors}}{\underbrace{1 \otimes \dots \otimes 1}} \otimes Z \otimes \underset{d-i-1 \text{ factors}}{\underbrace{1 \otimes \dots \otimes 1}} \in B^{\otimes d}. \end{align} $$

Moreover, for an endomorphism $\phi \in \operatorname {End}_K(B\otimes B)$ , set

(3.1.2) $$ \begin{align} \phi_i(b_1 \otimes \dots \otimes b_d) := b_1 \otimes \dots \otimes b_{i-1}\otimes \phi(b_{i} \otimes b_{i+1}) \otimes b_{i+2} \otimes \dots \otimes b_{d}. \end{align} $$

Definition 3.1.1 (Quantum wreath product).

Let B be an associative K-algebra, and let $d\in \mathbb {Z}_{\geq 2}$ . Let $Q = (R,S,\rho ,\sigma )$ be a choice of parameters with $R, S \in B \otimes B$ , and $\rho , \sigma \in \operatorname {End}_K(B\otimes B)$ such that $\sigma $ is an automorphism. The quantum wreath product is the associative K-algebra generated by the algebra $B^{\otimes d}$ and $H_1, \dots , H_{d-1}$ such that the following conditions hold, for $1\leq k \leq d-2, 1\leq i \leq d-1, |j-i|\geq 2$ :

(3.1.3) $$ \begin{align} &\text{(braid relations) } &H_k H_{k+ 1} H_k = H_{k+ 1} H_k H_{k + 1}, \quad H_i H_j = H_j H_i, \end{align} $$

(3.1.4) $$ \begin{align} &\text{(quadratic relations) } &H_i^2 = {S_i}H_i + {R_i}, \end{align} $$

(3.1.5) $$ \begin{align} &\text{(wreath relations) } & H_ib = \sigma_i(b)H_i + \rho_i(b) \quad ( b\in B^{\otimes d}). \end{align} $$

Oftentimes, we refer this algebra as $B \wr \mathcal {H}(d)$ , or $B \wr _Q \mathcal {H}(d)$ whenever it is convenient.

Here, ${R_i}$ is an element in $B^{\otimes d}$ , in contrast to those endomorphisms $\text {R}_{i,i+1}$ in the literature acting on the ith and $(i+1)$ th factors, in the context of the Yang–Baxter equations.

The quadratic and wreath relations determine the following local relations in $B\otimes B$ , in the sense that one can recover Equations (3.1.4)–(3.1.5) by embedding the local relation into the ith and $(i+1)$ th positions of $B^{\otimes d}$ as well as replacing the symbol H by $H_i$ :

(3.1.6) $$ \begin{align} H^2 = SH + R, \quad H (a\otimes b) = \sigma(a\otimes b) H + \rho(a \otimes b), \quad (a,b \in B). \end{align} $$

3.2 Necessary conditions

We first describe certain conditions on Q:

(3.2.1) $$ \begin{align} &\sigma(1\otimes 1) = 1\otimes 1, \quad \rho(1 \otimes1)=0, \end{align} $$

(3.2.2) $$ \begin{align} & \sigma(ab) = \sigma(a)\sigma(b), \quad \rho(ab) = \sigma(a)\rho(b) + \rho(a)b, \end{align} $$

(3.2.3) $$ \begin{align} & \sigma(S)S + \rho(S) + \sigma(R) = S^2+R, \quad \rho(R) + \sigma(S)R = SR, \end{align} $$

(3.2.4) $$ \begin{align} & r_S\sigma^2+\rho\sigma+\sigma\rho = {l}_{S}\sigma, \quad r_R\sigma^2+\rho^2 = {l}_S\rho + {l}_R, \end{align} $$

where ${l_X}, r_X$ for $X\in B\otimes B$ are K-endomorphisms defined by left and right multiplication in $B\otimes B$ by X, respectively. The next five conditions will only be necessary when $d\geq 3$ :

(3.2.5) $$ \begin{align} &\sigma_i\sigma_j\sigma_i = \sigma_j \sigma_i \sigma_j, \quad \rho_i\sigma_j\sigma_i = \sigma_j \sigma_i \rho_j, \end{align} $$

(3.2.6) $$ \begin{align} & \rho_i\sigma_j\rho_i = r_{S_j} \sigma_j \rho_i \sigma_j + \rho_j \rho_i \sigma_j + \sigma_j \rho_i \rho_j, \end{align} $$

(3.2.7) $$ \begin{align} &\rho_i\rho_j\rho_i + r_{R_i} \sigma_i \rho_j \sigma_i = \rho_j\rho_i\rho_j + r_{R_j} \sigma_j \rho_i \sigma_j, \end{align} $$

where $\{i,j\}= \{1,2\}$ , $r_X$ for $X\in B^{\otimes 3}$ is understood as right multiplication in $B^{\otimes 3}$ by X.

(3.2.8) $$ \begin{align} &S_i = \sigma_j\sigma_i(S_j), \quad R_i = \sigma_j\sigma_i(R_j), \quad \rho_j\sigma_i(S_j) = 0 = \rho_j\sigma_i(R_j), \end{align} $$

(3.2.9) $$ \begin{align} & \sigma_j\rho_i(S_j) S_j +\rho_j\rho_i(S_j) + \sigma_j\rho_i(R_j) = 0 = \rho_j\rho_i(R_j) + \sigma_j\rho_i(S_j) R_j, \end{align} $$

where $\{i,j\}= \{1,2\}$ .

Let $\{b_i \}_{i\in I}$ be a basis of B for some index set I, and let $H_w := H_{i_1} \dots H_{i_N} \in B\wr \mathcal {H}(d)$ for a reduced expression $w = s_{i_1} \dots s_{i_N}$ . Such an element $H_w$ is well defined due to the braid relations (3.1.3).

The conditions can be simplified when we put reasonable assumptions on Q. For example, if we assume that $\sigma $ is the flip map $a\otimes b \mapsto b\otimes a$ and that $\rho $ is the zero map, then Conditions (3.2.1)–(3.2.9) are equivalent to the following conditions:

(3.2.10) $$ \begin{align} R = \sigma(R), \quad (\sigma(S)-S)R = 0, \quad {r_S = l_S \sigma}. \end{align} $$

Note that the more involved conditions such as Equations (3.2.6)–(3.2.9) do not simplify much when $\rho \neq 0$ even if $\sigma $ is the flip map.

3.3 Structure theory

Now, we state the basis theorem regarding the sufficient and necessary conditions on Q such that $B \wr \mathcal {H}(d)$ affords desirable bases. This theorem will be proved in Section 5.

3.4

Consider a quantum wreath product $B \wr \mathcal {H}(d)$ . Assume additionally that B is an augmented algebra with counit $\epsilon : B \to K$ , R is invertible and that there is an $h_i \in K$ such that $h_i^2 = \epsilon (S) h_i + \epsilon (R)$ for each i. The space $B^{\otimes d}$ is also an augmented algebra, whose counit is also denoted by $\epsilon $ . It is natural to find the conditions on $\sigma $ and $\rho $ such that $B\wr \mathcal {H}(d)$ is an augmented algebra whose counit is given by

(3.4.1) $$ \begin{align} \epsilon : B\wr \mathcal{H}(d) \to K, \quad b_1 \otimes \dots \otimes b_d \mapsto \epsilon(b_1) \dots \epsilon(b_d), \quad H_i \mapsto h_i. \end{align} $$

The counit is well defined if and only if $\epsilon (H_i H_{i+1} H_i) = \epsilon (H_i H_{i+1} H_i)$ and $\epsilon (H_i b) = \epsilon (\sigma _i(b) H_{i} + \rho _i(b))$ for all i and $b\in B^{\otimes d}$ . Equivalently, $h_i$ ’s are all the same since R (and hence $h_i$ ’s) are invertible. We can write $h := h_i$ . Moreover,

(3.4.2) $$ \begin{align} h(\epsilon(b) - \epsilon\sigma(b)) = \epsilon\rho (b) \quad \text{for all } b \in B\otimes B. \end{align} $$

Hence, $B \wr \mathcal {H}(d)$ is augmented if $\epsilon \sigma = \epsilon $ and $\epsilon \rho = 0$ .

Note that Equation (3.4.2) can hold for more general $\sigma $ and $\rho $ . For examples, those satisfying $\sigma (b_1 \otimes b_2) = k(b_2\otimes b_1)$ for some unit $k\in K^{\times }$ , and $h(1-k) \epsilon = \epsilon \rho \in \mathrm {Hom}(B\otimes B, K)$ . Under the assumption (3.4.2), we can provide more information about the quantum wreath products and their quotients.

Proof. (a) Since $(B^{\otimes d})^+ \lhd B^{\otimes d}$ is an ideal, an arbitrary element in $A(B^{\otimes d})^+ = \langle H_i \rangle (B^{\otimes d})^+$ is of the form $\sum _{w}H_wb_w$ , where $\epsilon (b_w) =0$ for all w. Thus, it suffices to show that for any $b \in B^{\otimes d}$ with $\epsilon (b) =0$ , $H_ib \in (B^{\otimes d})^+A = (B^{\otimes d})^+ \langle H_j\rangle $ for all i.

Since $\sigma (b_1 \otimes b_2) \in Kb_2\otimes b_1$ , $\epsilon (\sigma _i(b))$ is a multiple of $\epsilon (b)=0$ . Thus,

(3.4.3) $$ \begin{align} H_ib = \sigma_i(b)H_i + \rho_i(b) \in (B^{\otimes d})^+ \langle H_j\rangle = (B^{\otimes d})^+A, \end{align} $$

where $\rho _i(b) \in (B^{\otimes d})^+$ since $\epsilon (\rho (b)) \in K \epsilon (b) =0$ for all $b\in (B^{\otimes d})^+$ , thanks to Equation (3.4.2).

(b) Since B is symmetric, $\beta : B\times B \to K, (a,b) \mapsto \mathop {\text {tr}}(ab)$ is a nondegenerate associative symmetric bilinear form. We claim that such a form for A is $\beta _A:A \times A \to K$ , given by

(3.4.4) $$ \begin{align} \beta_A(aH_x, bH_y) = \mathop{\text{tr}}(aH_x bH_y), \quad \text{where} \quad \mathop{\text{tr}}(bH_w) :=\begin{cases} \mathop{\text{tr}}(b) &\text{if } w=1; \\ 0 &\text{otherwise}. \end{cases} \end{align} $$

This trace map is well defined since A has the bases as in Theorem 3.3.1. It is not hard to verify that $\beta _A$ is nondegenerate and associative (for this to be true, it is crucial that R is invertible). It suffices to check that $\mathop {\text {tr}}(aH_x bH_y) = \mathop {\text {tr}}(bH_y aH_x )$ for all $a, b\in B^{\otimes d}, x,y \in \Sigma _d$ . By applying the wreath relation iteratively, we obtain $H_ya = \sigma _{y}(a) H_y$ , where $\sigma _{y}:= \sigma _{i_1} \cdots \sigma _{i_N}$ for a reduced expression $y = s_{i_1} \cdots s_{i_N} \in \Sigma _d$ , which is well defined as a consequence of Equation (3.2.5). Hence, for all $a, b \in B^{\otimes d}$ ,

(3.4.5) $$ \begin{align} \mathop{\text{tr}}(bH_yaH_x) =\mathop{\text{tr}}(b\sigma_y(a) H_yH_x), \quad \mathop{\text{tr}}(aH_x b H_y) =\mathop{\text{tr}}(a\sigma_x(b) H_xH_y). \end{align} $$

Both traces are zero unless $x=y^{-1}$ . In the case $x = y^{-1}$ , assume that $x = s_{i_1} \cdots s_{i_N}$ is a reduced expression. If we write $H_y H_x = \sum _{z\in \Sigma _d} c_z H_z$ , then

(3.4.6) $$ \begin{align} c_1 = (\sigma_{i_{N}}\cdots \sigma_{i_{2}})(R_{i_{1}})(\sigma_{i_{N}}\cdots \sigma_{i_{3}})(R_{i_{2}})\cdots \sigma_{i_{N}}(R_{i_{N-1}})R_{i_N}. \end{align} $$

On the other hand, if we write $H_x H_y = \sum _{z\in \Sigma _d} c^{\prime }_z H_z$ , then

(3.4.7) $$ \begin{align} c^{\prime}_1 = (\sigma_{i_{1}}\cdots \sigma_{i_{N-1}})(R_{i_{N}})\cdots \sigma_{i_{1}}(R_{i_{2}})R_{i_1}. \end{align} $$

Note that when $\rho = 0$ and R is invertible, Equation (3.2.3) become $\sigma (S) = S$ and $\sigma (R) = R$ . Since $\sigma _x = \sigma _{i_1} \cdots \sigma _{i_N}$ preserves the trace,

(3.4.8) $$ \begin{align} \begin{aligned} \mathop{\text{tr}}(b\sigma_y(a)H_x H_y) &= \mathop{\text{tr}}(b\sigma_y(a) (\sigma_{i_{N}}\cdots \sigma_{i_{2}})(R_{i_{1}})\cdots \sigma_{i_{N}}(R_{i_{N-1}})R_{i_N}) \\ &= \mathop{\text{tr}}(\sigma_x(b)a \sigma_{i_1}(R_{i_1}) (\sigma_{i_{1}}\sigma_{i_{2}})(R_{i_{2}})\cdots (\sigma_{i_{1}}\cdots \sigma_{i_{N}})(R_{i_{N}})) \\ &= \mathop{\text{tr}}(\sigma_x(b)a R_{i_1} \sigma_{i_{1}}(R_{i_{2}})\cdots (\sigma_{i_{1}}\cdots \sigma_{i_{N-1}})(R_{i_{N}})) = \mathop{\text{tr}}(a\sigma_x(b) H_x H_y), \end{aligned} \end{align} $$

and so the two traces agree, that is, A is a symmetric algebra.

4.1 Hu algebras

The reader is referred to Section 6 for the definition of the Hu algebra $ \mathcal {A}(m)$ . The algebra $\mathcal {A}(m)$ is essentially different from all other examples in Section 4. It is not covered by the theory of Rosso–Savage nor of Kleshchev–Muth due to its intriguing quadratic relation.

Assume that K is a field of characteristic not equal to 2, $f_{2m}(q)\neq 0$ , $B=\mathcal {H}_q(\Sigma _m)$ and the parameter $Q = (R,S,\rho ,\sigma )$ is given by

(4.1.1) $$ \begin{align} R = z_{m,m}, \quad S = 0, \quad \sigma: a\otimes b \mapsto b\otimes a, \quad \rho = 0. \end{align} $$

For this choice, almost all conditions (3.2.1)–(3.2.9) follow immediately (see Equation (3.2.10)). The only nontrivial condition, $\sigma (R) = R$ , follows the construction of $z_{m,m}$ in Section 6.2.

The algebra $\mathcal {H}_q(\Sigma _m) \wr \mathcal {H}(2)$ is generated by $T_x \otimes T_y\ (x, y \in \Sigma _m)$ and $H_1$ . Note that there are no braid relations for a single generator $H_1$ . The quadratic relation translates to that $H_1^2 = z_{m,m}$ , and the wreath relation becomes

(4.1.2) $$ \begin{align} H_1(T_x\otimes T_y) = (T_y \otimes T_x)H_1 \quad (x, y \in \Sigma_m). \end{align} $$

A dimension argument (by Proposition 6.2.2) shows that the Hu algebra $ \mathcal {A}(m)$ is canonically isomorphic to $\mathcal {H}_q(\Sigma _m) \wr \mathcal {H}(2)$ via the assignment below:

(4.1.3) $$ \begin{align} T_i \mapsto \begin{cases} T_i \otimes 1 &\text{if } 1\leq i \leq m-1; \\ 1\otimes T_{i-m} &\text{if } m+1 \leq i \leq 2m-1, \end{cases} \quad H_1(m) \mapsto H_1. \end{align} $$

4.2 (Degenerate) Affine Hecke algebras

See Appendix A.1 for definitions of variants of affine Hecke algebras. Now, we consider the case when the base algebra B is generated by X and that $\sigma : a\otimes b \mapsto b\otimes a$ is the flip map. Verifying relations for all these variants can be made easier via the Demazure operator $\partial $ , given by

(4.2.1) $$ \begin{align} \partial(X^a \otimes X^b) = \frac{X^a\otimes X^b - X^b \otimes X^a}{X\otimes 1 - 1\otimes X}. \end{align} $$

In particular, $\partial (X\otimes 1) = 1 = - \partial (1\otimes X)$ .

Proposition 4.2.1. Let $X \in B$ be an element not in the base ring K. Let $\sigma \in \operatorname {End}(B\otimes B)$ be the flip map, and let $\rho \in \operatorname {End}(B\otimes B)$ be such that Equation (3.2.2) holds. If $\beta := \rho (X\otimes 1)$ commutes with both $X\otimes 1$ and $1\otimes X$ , then the following holds for all $i,j \geq 0$ :

(4.2.2) $$ \begin{align} \rho(X^i \otimes X^j) = \partial(X^i\otimes X^j)\beta. \end{align} $$

Moreover, if $X^{-1} \in B$ , then Equation (4.2.2) also holds for $i,j \in \mathbb {Z}$ .

Proof. First, it follows from Equation (3.2.2) that

(4.2.3) $$ \begin{align} \rho(X \otimes X) = \rho((X\otimes 1)(1\otimes X)) = (1\otimes X) (\rho(1\otimes X)+ \beta). \end{align} $$

On the other hand, $\rho (X \otimes X)$ is also equal to $\rho ((1\otimes X)(X\otimes 1)) = (X\otimes 1) (\beta + \rho (1\otimes X))$ , and hence $\rho (1 \otimes X) = -\beta $ . An inductive argument shows that, for all $i \geq 1$ ,

(4.2.4) $$ \begin{align} \rho(X^i\otimes 1) = \beta \partial(X^i\otimes 1) = -\rho(1\otimes X^i), \end{align} $$

and hence,

(4.2.5) $$ \begin{align} \begin{aligned} \rho(X^i \otimes X^j) &= (1\otimes X^i) \rho(1\otimes X^j) + \rho(X^i \otimes 1)(1\otimes X^j) \\ &=\beta ((X^{i-1}\otimes X^j + \dots + 1\otimes X^{i+j-1}) - (X^{j-1}\otimes X^i + \dots + 1\otimes X^{i+j-1})) \\ &= \beta \partial(X^i \otimes X^j). \end{aligned} \end{align} $$

This first part is done. Assume from now on $X^{-1} \in B$ . We apply Equation (3.2.2) to both $\rho ((X\otimes 1)(1\otimes X^{-1}))$ and $\rho ((1\otimes X^{-1})(X\otimes 1))$ and then equate the two. We then obtain (4.2.2) for ${i=0, j<0}$ . Similarly, by applying Equation (3.2.2) to both $\rho ((X^{-1}\otimes 1)(1\otimes X))$ and $\rho ((1\otimes X)(X^{-1}\otimes 1))$ and then equating the two, we obtain Equation (4.2.2) for ${i<0, j=0}$ . The most general case is then obtained by applying Equation (3.2.2) to $\rho ((X^i \otimes 1)(1\otimes X^j))$ .

In other words, for the verification of the conditions appearing in Theorem 3.3.1 involving $\rho $ , one only need to check them on the generator $X\otimes 1$ as long as $\beta := \rho (X\otimes 1)$ commutes with both $X\otimes 1$ and $1\otimes X$ .

Now, for any $\mu = \sum _{i=1}^d \mu _i \epsilon _i \in \sum _i \mathbb {Z} \epsilon _i$ , write $X^\mu \equiv X^{\mu _1}\otimes \dots \otimes X^{\mu _d}$ , and let $s_j$ act on $\sum _i \mathbb {Z} \epsilon _i$ by place permutation on the jth and $(j+1)$ th positions. For an element $X\in B$ , for $1\leq j \leq d$ , set

(4.2.6) $$ \begin{align} X^{(j)} := \underset{j-1 \text{ factors}}{\underbrace{1 \otimes \dots \otimes 1}} \otimes X \otimes \underset{d-j \text{ factors}}{\underbrace{1 \otimes \dots \otimes 1}} \in B^{\otimes d}. \end{align} $$

Corollary 4.2.2 (Bernstein–Lusztig relations).

Retain the assumptions in Proposition 4.2.1. The following local equalities in $B \wr \mathcal {H}(d)$ hold in the sense of Equation (3.1.6):

(4.2.7) $$ \begin{align} H (X\otimes 1) H= (1\otimes X)R+ ((1\otimes X)S + \beta) H. \end{align} $$

(4.2.8) $$ \begin{align} H (X^i\otimes X^j) = (X^j\otimes X^i)H + \partial(X^i\otimes X^j)\beta. \end{align} $$

In particular, the following relations in $B \wr \mathcal {H}(d)$ hold, for $1\leq i \leq d-1$ , and for all $\mu $ ,

(4.2.9) $$ \begin{align} H_i X^\mu = X^{s_i(\mu)}H_i + \frac{X^\mu - X^{s_i(\mu)}}{X^{\epsilon_{i}}-X^{\epsilon_{i+1}}}\rho_i(X^{(i)}). \end{align} $$

4.2.1 Affine Hecke algebras of type A

Let $B = K[X^{\pm 1}]$ be the Laurent polynomials in X over a field K, $q \in K^{\times }$ , and let

(4.2.10) $$ \begin{align} \begin{aligned} &R = q(1\otimes1), \quad \sigma: {f\otimes g \mapsto g \otimes f, \quad (f,g\in B)}, \\ &S= (q-1)(1\otimes1), \quad \rho: X^a \otimes X^b \mapsto -(q-1)\partial(X^a \otimes X^b)(1\otimes X). \end{aligned} \end{align} $$

The extended affine Hecke algebra $\mathcal {H}^{\mathrm {ext}}_q(\Sigma _d)$ can be realized via the quantum wreath product $K[X^{\pm 1}] \wr \mathcal {H}(d)$ under the identification, for $\lambda = (\lambda _1, \dots , \lambda _d)$ :

(4.2.11) $$ \begin{align} T_i \mapsto H_i, \quad Y^\lambda \mapsto X^\lambda \equiv X^{\lambda_1} \otimes \dots \otimes X^{\lambda_d}. \end{align} $$

Since $\beta = -(q-1) 1\otimes X$ commutes with either $X\otimes 1$ or $1\otimes X$ , Corollary 4.2.2 follows. Therefore, the wreath relations (3.1.5) implies the Bernstein–Lusztig relations (A.1.1) since

$$ \begin{align*}\frac{X^\mu - X^{s_i(\mu)}}{X^{\epsilon_{i}}-X^{\epsilon_{i+1}}}\beta = (q-1) \frac{X^\mu - X^{s_i(\mu)}}{1-X^{\epsilon_i - \epsilon_{i+1}}}.\end{align*} $$

It is easy to see that the Bernstein–Lusztig relation implies the wreath relation, and hence, they are equivalent.

Since it is well known (see [Reference LusztigLu89]) that $\mathcal {H}^{\mathrm {ext}}_q(\Sigma _d)$ admits a basis of the form $\{Y^\mu T_w ~|~ \mu \in P, w\in \Sigma _d\}$ , conditions (3.2.1)–(3.2.9) hold as long as Theorem 3.3.1 is proved. On the other hand, one can give another proof of Lusztig’s basis theorem by a tedious but elementary verification of conditions (3.2.1)–(3.2.9).

Note that by this choice of $R, S$ , we have $H(X\otimes 1)H = q(1\otimes X)$ from Corollary 4.2.2, or equivalently, $H_i X^{(i)} H_i = qX^{(i+1)}$ for all i. If we choose $R = 1\otimes 1, S = (q^{-1} - q)(1\otimes 1)$ instead, we would obtain $H_i X^{(i)} H_i = X^{(i+1)}$ for all i.

The affine Hecke algebra $\mathcal {H}_q(\Sigma _d^{\mathrm {aff}})$ is then realized as the subalgebra

(4.2.12) $$ \begin{align} \mathcal{H}_q(\Sigma_d^{\mathrm{aff}}) \equiv \{X^\lambda H_w ~|~ \Sigma_i \lambda_i = 0, w\in \Sigma_d\}. \end{align} $$

In particular, the affine generator $T_0\in \mathcal {H}_q(\Sigma _d^{\mathrm {aff}})$ can be identified as

(4.2.13) $$ \begin{align} T_0 \equiv (X \otimes 1^{\otimes d-2} \otimes X^{-1}) H_{d-1} \dots H_2 H_1 H_2 \dots H_{d-1}. \end{align} $$

4.2.2 Ariki–Koike algebras

Let $q_1, \dots , q_m\in K$ , and let $Q= (R, S, \rho , \sigma )$ be as in (4.2.10). The Ariki–Koike algebra is the following cyclotomic quotient of the quantum wreath product

(4.2.14) $$ \begin{align} \mathcal{H}_{q,q_1, \dots, q_m}(C_m\wr \Sigma_d) = \frac{K[X^{\pm1}] \wr \mathcal{H}(d)}{\langle (X-q_1) \dots (X-q_m)\otimes 1^{\otimes d-1}\rangle}. \end{align} $$

In particular, the Jucys–Murphy elements $L_i$ ’s are identified as below:

(4.2.15) $$ \begin{align} L_1 = T_0 \equiv \overline{X \otimes 1^{\otimes d-1}} =: \overline{X^{(1)}}, \quad L_{i+1} = {q^{-1}} T_i L_i T_i \equiv \overline{X^{(i+1)}} \quad \text{for} \quad 0 \leq i \leq d-1. \end{align} $$

Therefore, the well-known surjection $\mathcal {H}_q^{\mathrm {ext}}(\Sigma _d) \to \mathcal {H}_{q,q_1, \dots , q_m}(C_m\wr \Sigma _d), Y^{\epsilon _i} \mapsto L_i$ is the canonical map

(4.2.16) $$ \begin{align} K[X^{\pm1}] \wr \mathcal{H}(d) \to \frac{K[X^{\pm1}] \wr \mathcal{H}(d)}{\langle (X-q_1) \dots (X-q_m)\otimes 1^{\otimes d-1}\rangle}, \quad X^\mu H_w \mapsto \overline{X^\mu H_w} \quad (\mu \in \mathbb{Z}^d, w\in \Sigma_d). \end{align} $$

4.2.3 Affine Hecke algebras of type B/C

The (nonextended) affine Hecke algebra $\mathcal {H}(C^{\mathrm {aff}}_d)$ of type C is the Hecke algebra $\mathcal {H}(C^{\mathrm {aff}}_d) = \langle T_0, \dots , T_d\rangle $ for the Coxeter group of type $C^{\mathrm {aff}}_d$ , in which $T_0$ corresponds to the type C node in the Dynkin diagram, while $T_d$ corresponds to the affine node. Let $B = K[X^{\pm 1}, Y^{\pm 1}]$ , $q, \xi , \eta \in K^{\times }$ , and let

(4.2.17) $$ \begin{align} \begin{aligned} &R = (1\otimes1), \quad \sigma: f\otimes g \mapsto g \otimes f, \quad (f,g\in B), \\ &S= (q^{-1}-q)(1\otimes1), \quad \rho: \begin{cases} X \otimes 1 \mapsto (q-q^{-1})(1\otimes X); \\ Y \otimes 1 \mapsto (q^{-1}-q)(Y\otimes 1). \end{cases} \end{aligned} \end{align} $$

Then

(4.2.18) $$ \begin{align} \mathcal{H}(C^{\mathrm{aff}}_d) = \frac{K[X^{\pm1}, Y^{\pm1}] \wr \mathcal{H}(d)}{\langle (X+\xi)(X-\xi^{-1})\otimes 1^{\otimes d-1}, 1^{\otimes d-1} \otimes (Y+\eta)(Y-\eta^{-1})\rangle}, \end{align} $$

under the identification

(4.2.19) $$ \begin{align} T_0 \mapsto \overline{X^{(1)}}, \quad T_i \mapsto \overline{H_i} \quad (1\leq i \leq d-1), \quad T_d \mapsto \overline{Y^{(d)}}. \end{align} $$

On the other hand, the extended affine Hecke algebra $\mathcal {H}^{\mathrm {ext}}(B_d)$ of type B is generated by $T_0$ , $T_1$ , $\dots $ , $T_{d-1}$ , $Y^\lambda (\lambda \in P(B^\vee _d))$ , where $T_0$ corresponds to the type B node in the Dynkin diagram. It is possible to identify $\mathcal {H}^{\mathrm {ext}}(B_d)$ as a subalgebra of $K[X^{\pm 1}] \wr \mathcal {H}(2d)$ (with respect to Equation (4.2.10)) generated by

(4.2.20) $$ \begin{align} \begin{aligned} &X^{\lambda_1} \otimes \dots \otimes X^{\lambda_{2d}} \quad ( \lambda_i = - \lambda_{2d+1-i} \text{ for all }i), \\ &H_d, \quad H_{d+1}H_{d-1},\quad \dots,\quad H_1H_{2d-1}. \end{aligned} \end{align} $$

4.2.4 Degenerate affine Hecke algebras

Let $\mathcal {H}^{\mathrm {deg}}(d)$ be the degenerate affine Hecke algebra of type A, generated by $s_1, \dots , s_{d-1}, x_i, \dots , x_d$ (see Appendix A.1). The $s_i$ ’s and $x_j$ ’s interact by the cross relations (A.2.2). Let $B = K[X]$ be the polynomial ring. We choose

(4.2.21) $$ \begin{align} R = 1\otimes 1, \quad S = 0, \quad \sigma: X^a\otimes X^b \mapsto X^b\otimes X^a, \quad \rho: X^a \otimes X^b \mapsto -\partial(X^a \otimes X^b). \end{align} $$

Then $\mathcal {H}^{\mathrm {deg}}(d) = K[X] \wr \mathcal {H}(d)$ under the identification

(4.2.22) $$ \begin{align} s_i \mapsto H_i, \quad x_j \mapsto X^{(j)}. \end{align} $$

For $\lambda = \sum _i \lambda _i \epsilon \in P^+ := \sum _{i=1}^d \mathbb {Z}_+\epsilon _i$ , we set $x^\lambda := x_1^{\lambda _1} \dots x_d^{\lambda _d}$ . Now, $\beta = - 1$ commutes with either $X\otimes 1$ or $1\otimes X$ , and hence Corollary 4.2.2 follows. Similar to the case for affine Hecke algebras, our wreath relations (3.1.5) are equivalent to the Bernstein–Lusztig–type relations below, for $\lambda \in P^+, 1\leq i \leq d-1$ :

(4.2.23) $$ \begin{align} s_i x^\lambda= x^{s_i(\lambda)} s_i - \frac{x^\lambda - x^{s_i(\lambda)}}{x_i - x_{i+1}}. \end{align} $$

Since it is well known (see [Reference LusztigLu89]) that $\mathcal {H}^{\mathrm {deg}}(d)$ admits a basis of the form $\{x^\lambda w ~|~ \lambda \in P^+, w\in \Sigma _d\}$ , conditions (3.2.1)–(3.2.9) hold as long as Theorem 3.3.1 is proved.

Kostant–Kumar’s nil Hecke rings (see Appendix A.3) can be realized similarly by setting $R=0$ , instead.

4.2.5 Hecke-type categories

It is known that the Nil Hecke rings describe the endomorphism spaces in the Khovanov–Lauda–Rouquier (KLR) algebras. This generalizes to Elias’ Hecke-type category, whose endomorphism spaces (of rank d) can be regarded as an algebra generated by $H_1, \dots , H_{d-1}$ and $B^{\otimes d}$ (where B is a commutative ring; in contrast to our definition in which B is an arbitrary associative algebra). Their algebras satisfy our quadratic relations (3.1.4) as well as wreath relations (3.1.5), while the braid relations are relaxed as follows:

(4.2.24) $$ \begin{align} H_{i} H_{i+1} H_i \in K^{\times} H_{i+1} H_iH_{i+1} +\sum_{x < s_i s_{i+1} s_i} (B\otimes B\otimes B) H_x. \end{align} $$

The key point is that our quantum wreath products is a special case of Elias’ construction when the base algebra B is commutative. However, in our main example, the Hu algebra, the base algebra $B= \mathcal {H}_q(\Sigma _m)$ is far from commutative.

4.3 Rosso–Savage algebras

Let $z\in K$ , and let B be a certain Frobenius K-algebra with a Nakayama automorphism $\psi :B\to B$ , a trace map $\text {tr}:B\to K$ , a basis I and its dual basis $\{b^\vee ~|~b\in I\}$ with respect to $\text {tr}$ . Rosso and Savage introduced in [Reference SavageSa20, Reference Rosso and SavageRS20] the Frobenius Hecke algebra $H_d(B,z)$ , its affinization $H^{\mathrm {aff}}_d(B,z)$ and the affine wreath product algebra $\mathcal {A}_d(B)$ which generalize the Hecke algebra, the affine Hecke algebra and the degenerate affine Hecke algebra, respectively (see Appendix A.5 for details).

We have the following identifications as quantum wreath products:

(4.3.1) $$ \begin{align} H_d(B,z) = B\wr \mathcal{H}(d), \quad H^{\mathrm{aff}}_d(B,z) = (B\hat{\otimes} K[X^{\pm1}]) \wr \mathcal{H}(d), \quad \mathcal{A}_d(B) = (B\hat{\otimes} K[X]) \wr \mathcal{H}(d), \end{align} $$

where $\hat {\otimes }$ denotes the twisted tensor product given by ${(b\otimes 1)X^{\pm 1} = X^{\pm 1}(\psi ^{\pm 1}(b) \otimes 1)}$ for $b\in B$ . Here, $R = 1\otimes 1$ and $\sigma (a\otimes b) = b\otimes a$ for all three cases. For $H_d(B,z)$ , the other parameters are

(4.3.2) $$ \begin{align} S=z \sum_{b \in I} (b\otimes b^\vee) \in B\otimes B, \quad \rho = 0. \end{align} $$

For $H^{\mathrm {aff}}_d(B,z)$ , one has

(4.3.3) $$ \begin{align} S=z\sum_{b \in I} (b\otimes b^\vee) \in B\otimes B, \quad \rho: \begin{cases} b_1\otimes b_2 \mapsto 0 &(b_i \in B); \\ X \otimes 1 \mapsto -(1\otimes X) z \sum_{b \in I} (b\otimes b^\vee) \in B\otimes B. \end{cases} \end{align} $$

For $\mathcal {A}_d(B)$ , one has

(4.3.4) $$ \begin{align} S=0, \quad \rho: \begin{cases} b_1\otimes b_2 \mapsto 0 &( b_i \in B); \\ X \otimes 1 \mapsto - \sum_{b \in I} (b\otimes b^\vee) \in B\otimes B. \end{cases} \end{align} $$

In particular, these algebras include the Yokonuma–Hecke algebra $Y_{m,d} = KC_m \wr \mathcal {H}(d)$ , its affinization, Wan–Wang’s [Reference Wan and WangWW08] wreath Hecke algebra $\mathcal {H}^{\mathrm {deg}}(G\wr \Sigma _d) = K[X]G \wr \mathcal {H}(d)$ , Evseev–Kleshchev’s super wreath product algebra.

In [Reference Savage and StuartSS21], Savage and Stuart also considered variants of these algebra by replacing the quadratic relation with $H_i^2=0$ . By a similar argument, one can show that their Frobenius nil Coxeter algebras are also special cases of ours. Their Frobenius nil Hecke algebras can also be included as long as the base algebra is purely even.

4.4 Affine zigzag algebras

Readers are referred to Section A.6 for the definition of the zigzag algebra $Z_K(\Gamma )$ , its affinization $Z^{\mathrm {aff}}_d(\Gamma )$ over a simply-laced connected Dynkin diagram $\Gamma $ . Let $B = Z_{K[X]}(\Gamma )$ be the zigzag algebra over the polynomial ring $K[X]$ . Let $R = 1\otimes 1, S = 0, \sigma : a\otimes b \mapsto b\otimes a$ , and let $\rho : B\otimes B \to B \otimes B$ be determined by, for $\gamma _i \in Z_K, i \neq j \in I$ :

(4.4.1) $$ \begin{align} \rho(\gamma_1 \otimes \gamma_2) = 0, \quad \rho(Xe_i \otimes e_i) = (c_1+c_2) (e_i \otimes e_i), \quad \rho(Xe_i \otimes e_j) = a_{j,i} \otimes a_{i,j}. \end{align} $$

In other words, we have the following local relations, for $\gamma _i \in Z_K, i \neq j \in I$ :

(4.4.2) $$ \begin{align} \begin{aligned} &H(\gamma_1\otimes \gamma_2) = (\gamma_2 \otimes \gamma_1) H, \\ &H(Xe_i \otimes e_i) = (e_i \otimes Xe_i)H + (c_1+c_2) (e_i \otimes e_i), \quad H(Xe_i \otimes e_j) = (e_j \otimes Xe_i)H + a_{j,i} \otimes a_{i,j}. \end{aligned} \end{align} $$

Then $Z_d^{\mathrm {aff}}(\Gamma ) = Z_{K[X]} \wr \mathcal {H}(d)$ under the identification with $T_i \mapsto H_i$ , $z_j \mapsto X^{(j)}$ .

5.1

We start with a left $B^{\otimes d}$ -module $V := B^{\otimes d} \otimes K \Sigma _d$ , on which $B^{\otimes d}$ acts by left multiplication. Let $A := B\wr \mathcal {H}(d)$ . Our goal is to show that, under suitable conditions, V can be given a right A-module structure, and the assignment $a\otimes w \mapsto a H_w$ for $a\in B^{\otimes d}, w\in \Sigma _d$ defines a right A-module isomorphism $V \to A$ . Note that V admits a filtration $(V^\ell )_{\ell \in \mathbb {N}}$ of $B^{\otimes d}$ -modules given by

(5.1.1) $$ \begin{align} V^\ell := \bigoplus_{\ell(w) \leq \ell} B^{\otimes d} \otimes K w. \end{align} $$

We fix a reduced expression $r(w)$ for all $w\in \Sigma _d$ that is compatible with others, that is, $r(1) = 1$ and for every $w \in \Sigma _d \setminus \{1\}$ , there exists a unique $s_i$ such that $r(w) = r(w s_i) s_i$ as reduced expressions. Existence of such a choice follows from an inductive argument.

In order to eliminate the left-versus-right issue, in Section 5, we use the following postfix notation such that the juxtaposition of maps means the reversed composition of maps, that is, for any two maps $f:M_1 \to M_2, g:M_2 \to M_3$ , it is understood that

(5.1.2) $$ \begin{align} fg := g \circ f : M_1 \to M_3, \quad \text{and} \quad (m)\cdot fg := (g\circ f)(m) \in M_3 \quad \text{for all} \quad m\in M_1, \end{align} $$

where $m\cdot f = f(m)$ is the postfix evaluation of f at m. In particular, for endomorphisms $f, g \in \operatorname {End}(M)$ , by the juxtaposition $fg$ we mean the the multiplication in the opposite ring $\operatorname {End}(M_1)^{\text {op}}$ .

Next, we define (left) $B^{\otimes d}$ -module homomorphisms $f^{(\ell )}_a = f^{(\ell )}_a(r) \in \operatorname {End}(V^\ell )$ , and $T_i^{(\ell )} = T_i^{(\ell )}(r):V^\ell \to V^{\ell +1}$ , for all $a\in B^{\otimes d}, 1\leq i \leq d-1, \ell \geq 0$ in a recursive manner:

(5.1.3) $$ \begin{align} (1^{\otimes d}\otimes w) \cdot f_{a}^{(\ell)} := \begin{cases} a \otimes 1, & \text{if } w = 1, \\ (1^{\otimes d} \otimes w s_i) \cdot ( f_{\sigma_i(a)}^{(\ell - 1)} T_{i}^{(\ell - 1)} + f_{\rho_i(a)}^{(\ell - 1)}), & \text{if } r(w) = r(w s_i) s_i, \\ \end{cases} \end{align} $$

(5.1.4) $$ \begin{align} (1^{\otimes d} \otimes w) \cdot T_i^{(\ell)} := \begin{cases} 1^{\otimes d}\otimes ws_i , &\text{if } \ell(w) < \ell(w s_i), \\ (1^{\otimes d}\otimes w s_i) \cdot (f_{S_i}^{(\ell-1)} T_i^{(\ell-1)} + f_{R_i}^{(\ell-1)}), &\text{if } \ell(w)> \ell(w s_i), \end{cases} \end{align} $$

for all w with length $\ell (w) \leq \ell $ . Note that the recursion does terminate since $T_i^{(0)}$ and $f_a^{(0)}$ are given by $(b \otimes 1) \cdot T_i^{(0)} = b\otimes s_i$ and $(b\otimes 1) \cdot f_{a}^{(0)} = (ba)\otimes 1$ for any $b\in B^{\otimes d}$ .

Proof. We prove this by an induction on j. The base case $f_a^{(\ell )}|_{V^0} = f_a^{(0)}$ and $T_i^{(\ell )}|_{V^0} = T_i^{(0)}$ follow from Equations (5.1.3) and (5.1.4), respectively. For the inductive case, pick any $b\otimes x \in V^j$ with $x \neq 1$ (i.e., $\ell (x) \leq j \leq \ell $ ). There is an $s_k$ such that $r(x) = r(w)s_k$ , and thus $b\otimes w \in V^{j-1}$ . Then

(5.1.5) $$ \begin{align} \begin{array}{ll} (b\otimes x) \cdot f_a^{(\ell)} = (b\otimes w) \cdot (f_{\sigma_k(a)}^{(\ell-1)}T_k^{(\ell-1)} + f_{\rho_k(a)}^{(\ell-1)}) &\text{by Equation}~({5.1.3}) \\ \quad =(b\otimes w) \cdot (f_{\sigma_k(a)}^{(j-1)}T_k^{(j-1)} + f_{\rho_k(a)}^{(j-1)}) &\text{by the induction hypothesis} \\ \quad = (b\otimes x) \cdot f_a^{(j)}. &\text{by Equation}~({5.1.3}) \end{array} \end{align} $$

The proof of that $T_i^{(\ell )}|_{V^j} = T_i^{(j)}$ for all $j \leq \ell , 1\leq i \leq d-1$ splits into two cases. The first case $\ell (x)> \ell (xs_i)$ follows from the fact that

(5.1.6) $$ \begin{align} \begin{array}{ll} (b\otimes x)\cdot T_i^{(\ell)} = (b\otimes xs_i)\cdot ( f_{S_i}^{(\ell-1)}T_i^{(\ell-1)} + f_{R_i}^{(\ell-1)}) &\text{by Equation} ({5.1.4}) \\ \quad = (b\otimes xs_i) \cdot (f_{S_i}^{(j-1)}T_i^{(j-1)} + f_{R_i}^{(j-1)}) &\text{by the induction hypothesis} \\ \quad = (b\otimes x) \cdot T_i^{(j)}. &\text{by Equation}~({5.1.4}) \end{array} \end{align} $$

The other case $\ell (x) < \ell (xs_i)$ follows from the fact that $(b\otimes x)\cdot T_i^{(\ell )} = b\otimes xs_i = (b\otimes x) \cdot T_i^{(j)}$ , thanks to Equation (5.1.4).

Let $f_a = f_a(r), T_i = T_i(r) \in \operatorname {End}_{B^{\otimes d}}(V)$ be given by $f_a|_{V^\ell } = f_a^{(\ell )}$ and $T_i|_{V^\ell } = T_i^{(\ell )}$ for all $\ell $ . Thanks to Proposition 5.1.1, these maps are well defined, and it leads to the following proposition from Equations (5.1.3)–(5.1.4).

Proposition 5.1.2. The following statements hold for all $w\in \Sigma _d, a\in B^{\otimes d}, 1\leq i \leq d-1$ :

(5.1.7) $$ \begin{align} & (1^{\otimes d}\otimes w) \cdot f_{a} := \begin{cases} a \otimes 1, & \text{if } w = 1, \\ (1^{\otimes d} \otimes w s_i) \cdot ( f_{\sigma_i(a)} T_{i} + f_{\rho_i(a)}), & \text{if } r(w) = r(w s_i) s_i. \\ \end{cases} \end{align} $$

(5.1.8) $$ \begin{align} & ( 1^{\otimes d}\otimes w) \cdot T_i = \begin{cases} 1^{\otimes d}\otimes ws_i , &\text{if } \ell(w) < \ell(ws_i), \\ (1^{\otimes d}\otimes w s_i) \cdot (f_{S_i} T_i + f_{R_i}), &\text{if } \ell(w)> \ell(ws_i). \\ \end{cases}\quad\qquad \end{align} $$

We remark that one cannot define $f_a(r)$ and $T_i(r)$ directly using Equations (5.1.7)–(5.1.8) since a circular definition will occur. Later, we will show that these maps are actually independent of r.

5.2 Grand loop argument

In the following, we present a key ingredient towards the proof of Theorem 3.3.1 that has a similar flavor as Kashiwara’s grand loop argument, that is, we will prove intermediate statements below based on an induction on $\ell $ :

(W[ℓ]) $$\begin{align} & T_i f_a = f_{\sigma_i(a)} T_i + f_{\rho_i(a)} \in \mathrm{Hom}_{B^{\otimes d}}(V^\ell, V) \text{ for all } a\in B^{\otimes d}, 1\leq i \leq d-1,\end{align} $$

(M[ℓ]) $$ \begin{align} & f_a f_b = f_{a b} \in \mathrm{Hom}_{B^{\otimes d}}(V^\ell, V) \text{ for all } a,b \in B^{\otimes d}, \end{align} $$

(Q[ℓ]) $$ \begin{align} & T_i T_i = f_{S_i} T_i + f_{R_i} \in \mathrm{Hom}_{B^{\otimes d}}(V^\ell, V) \text{ for all }1\leq i \leq d-1, \end{align} $$

(B2[ℓ]) $$ \begin{align} & T_i T_j = T_j {T_i} \in \mathrm{Hom}_{B^{\otimes d}}(V^\ell, V) \text{ for all } |i- j|>1, \end{align} $$

(B3[ℓ]) $$ \begin{align} & T_i T_j T_i = T_j T_i T_j \in \mathrm{Hom}_{B^{\otimes d}}(V^\ell, V) \text{ for all }|i- j| =1, \end{align} $$

(R[ℓ]) $$ \begin{align} & f_a \in \mathrm{Hom}_{B^{\otimes d}}(V^\ell, V) \text{ does not depend on }r. \end{align} $$

Note that the Condition $R[\ell ]$ implies that $T_i \in \mathrm {Hom}_{B^{\otimes d}}(V^\ell , V)$ does not depend on r as well.

Proposition 5.2.1. Assume that Equations (3.2.1)–(3.2.4) hold, and Equations (3.2.5)–(3.2.9) hold additionally if $d\geq 3$ .

$$\begin{align*}\begin{array}{ll} (W) &\text{Condition }W[\ell]\text{ follows from }R[\ell + 1], M[\ell - 1], W[\ell - 1]\text{ and }Q[\ell-1]. \\ (M) &\text{Condition }M[\ell]\text{ follows from }W[\ell - 1]\text{ and }M[\ell - 1]. \\ (Q) &\text{Condition }Q[\ell]\text{ follows from combining }Q[\ell - 1], M[\ell - 1]\text{ and }W[\ell - 1]. \\ (B2) &\text{Condition }B_2[\ell]\text{ follows from combining }B_2[\ell - 1], Q[\ell], W[\ell - 1]. \\ (B3) &\text{Condition }B_3[\ell]\text{ follows from }B_3[\ell - 1], M[\ell - 1], Q[\ell + 1]\text{ and }W[\ell]. \\ (R) &\text{Condition }R[\ell]\text{ follows from combining }R[\ell - 1], B_2[\ell - 2], B_3[\ell - 3]\text{ and }Q[\ell - 3]. \\ \end{array}\end{align*}$$

Proof. The base cases can be verified directly. For the inductive step, we assume that $W[i-2], M[i-1], Q[i-1], B_2[i-2], B_3[i-3], R[i-1]$ hold for all $i \leq \ell $ . Therefore, for $n \in \{2,3\}$ ,

$$\begin{align*}\begin{array}{lllllll}&W[\ell-2],&M[\ell-1],&Q[\ell-1],&B_n[\ell-n],&R[\ell-1]& \text{hold by inductive hypothesis,}\\\Rightarrow&W[\ell-2],&M[\ell-1],&Q[\ell-1],&B_n[\ell-n],&R[\ell]& \text{hold by Proposition}~{5.2.1}\ \text{(R)},\\\Rightarrow&W[\ell-1],&M[\ell-1],&Q[\ell-1],&B_n[\ell-n],&R[\ell]& \text{hold by Proposition}~{5.2.1}\ \text{(W)},\\\Rightarrow&W[\ell-1],&M[\ell],&Q[\ell-1],&B_n[\ell-n],&R[\ell]& \text{hold by Proposition}~{5.2.1}\ \text{(M)},\\\Rightarrow&W[\ell-1],&M[\ell],&Q[\ell],&B_n[\ell-n],&R[\ell]& \text{hold by Proposition}~{5.2.1}\ \text{(Q)},\\\Rightarrow&W[\ell-1],&M[\ell],&Q[\ell],&B_n[\ell+1-n],&R[\ell]& \text{hold by Proposition}~{5.2.1}\ \text{(Bn)}.\end{array} \end{align*}$$

The proof concludes by proceeding inductively.

We remark that the proof of Lemma 5.2.2 work perfectly even for the degenerate case when $d=2$ . Since there is a unique choice of r for $\Sigma _2$ , $R[\ell ]$ holds for all $\ell $ . On the other hand, $B_n[\ell ]$ are vacuous statements and hence always are true. The remaining conditions are verified by using the induction above. The grand loop degenerates to a loop that only involves $W[i]$ ’s, $M[i]$ ’s and $Q[i]$ ’s. The implications therein are given by Proposition 5.2.1 (M), (Q) and (W), which in fact only require Equations (3.2.1)–(3.2.4).

5.3 Proof of the basis Theorem 3.3.1

Proof. First, we prove that (a) is equivalent to (b). One direction is already taken care of by Proposition 3.2.1. For the other direction, we will give V a right A-module structure, and prove that the left $B^{\otimes d}$ -module homomorphism $\phi : V \to A$ , $a \otimes w \mapsto a H_w$ is a right A-module isomorphism.

Step 1: We first show that $\phi $ is onto. Note first an arbitrary element $x\in A$ is a sum of elements of the form $a_1 \cdots a_n$ for some $n=n(x)$ and each $a_j$ is either a generator $H_i$ or an element in $B^{\otimes d}$ . By applying the wreath, quadratic and braid relations iteratively, x becomes a sum of elements of the form

(5.3.1) $$ \begin{align} b H_{i_1} \cdots H_{i_N}, \quad b \in B^{\otimes d}, \quad N \leq n, \quad s_{i_1} \cdots s_{i_N} \text{ is a reduced expression}, \end{align} $$

and thus $\phi $ is onto.

Step 2: We show that $\phi $ is an isomorphism. Thanks to Lemma 5.2.2, $W[i], M[i], Q[i], B_2[i], B_3[i]$ and $R[i]$ hold for all i. Hence, the assignment $a \mapsto f_a, H_w \mapsto T_w$ induces an algebra homomorphism $\Phi : A \to \operatorname {End}_{B^{\otimes d}}(V)^{\text {op}}$ , where $T_w := T_{i_1} \cdots T_{i_N}$ if $w = s_{i_1} \cdots s_{i_N}$ is reduced. Using this map, we obtain a right A-module structure of V.

Note that any element in V is of the form $v = \sum _{w\in \Sigma _d} b_w \otimes w$ for some $b_w= b_w(v) \in B^{\otimes d}$ . It suffices to show that if $\sum _{w\in \Sigma _d} b_w H_w = 0$ , then $\sum _{w\in \Sigma _d} b_w\otimes w = 0$ . Indeed,

(5.3.2) $$ \begin{align} \sum_{w\in\Sigma_d} b_w\otimes w &= \sum_{w\in\Sigma_d} (b_w\otimes 1)\cdot T_w = \sum_{w\in\Sigma_d} (1^{\otimes d}\otimes 1)\cdot f_{b_w} T_w \notag \\ &= (1^{\otimes d}\otimes 1)\cdot \sum_{w\in\Sigma_d} f_{b_w} T_w = (1^{\otimes d}\otimes 1)\cdot \Phi(\textstyle\sum_{w\in\Sigma_d} b_w H_w) = 0. \end{align} $$

Therefore, (a) and (b) are equivalent.

Next, we show that (b) and (c) are equivalent. For each $w\in \Sigma _d$ , let $\underline {w} = (s_{i_1}, \dots , s_{i_n})$ be a fixed reduced expression of w, and set $\sigma _{\underline {w}} := \sigma _{i_1} \circ \dots \circ \sigma _{i_n}$ .

By applying the wreath, quadratic and braid relations iteratively to a typical element $H_w b \in A$ , one obtains

(5.3.3) $$ \begin{align} H_w b \in \sigma_{\underline{w}}(b) H_w + \sum_{x < w} B^{\otimes d} H_x, \end{align} $$

with respect to the Bruhat order on $\Sigma _d$ . By substituting b for $\sigma _{\underline {w}}^{-1}(b)$ , Equation (5.3.3) becomes

(5.3.4) $$ \begin{align} b H_w \in H_w \sigma_{\underline{w}}^{-1}(b) + \sum_{x < w} B^{\otimes d} H_x \subseteq H_w \sigma_{\underline{w}}^{-1}(b) + \sum_{x < w} H_xB^{\otimes d}, \end{align} $$

where the latter inclusion follows applying the former part of (5.3.4) to every $B^{\otimes d} H_x$ , iteratively. In other words, the transition matrices are invertible since every $\sigma _{\underline {w}}$ is an automorphism. Thus, (b) and (c) are equivalent.

5.4 Elias’ basis Theorem

In the study of Hecke-type categories (see Section 4.2.5), the base algebra B is always a commutative ring. In our setup, the base algebra B can be an associative algebra. A fundamental question is to determine whether a diagrammatic algebra generated by crossings and dots with relations given by resolving crossings has the right size. In [Reference EliasE22], Elias proved a Bergman diamond lemma for the endomorphism algebras (see Section 4.2.5) regarding sufficient and necessary conditions in terms of (conceptual) resolvable ambiguities, which corresponds to the very first step of our proof of Theorem 3.3.1.

In particular, our conditions (3.2.1)–(3.2.9) describe explicitly the requirements on the choice of parameters in order to resolve the minimal set of ambiguities given in [Reference EliasE22, (5.1), (5.11)], except for a couple of them that do not appear in our setup since we require the braid relations to hold in our case.

6.1 The construction of the element $h_{m}$

Let $d=2m$ . In order to quantize Equation (2.1.6), consider the type B Hecke algebra $\mathcal {H}^{\mathrm {B}} = \mathcal {H}_{(1,q)}(W(\mathrm {B}_{2m}))$ of unequal parameters $(1,q)$ . One can realize the Hecke algebras for $\Sigma _m \times \Sigma _m$ and for $\Sigma _{2m}$ as subalgebras of $\mathcal {H}^{\mathrm {B}}$ via the generating sets $\{T_1, \dots , T_{m-1}, T_{m+1}, \dots T_{d-1}\}$ and $\{T_1, \dots , T_{d-1}\}$ , respectively. It is well known that $T_w = T_{i_1} \cdots T_{i_r} \in \mathcal {H}^{\mathrm {B}}$ is well defined if $w = s_{i_1} \cdots s_{i_r} \in W(B_d)$ is a reduced expression. The tower of groups in Equation (2.1.6) can be recovered by taking the specialization $q=1$ from the following tower of algebras:

(6.1.1) $$ \begin{align} \mathcal{H}_q(\Sigma_m \times \Sigma_m) \subseteq \mathcal{A}(m) \subseteq \mathcal{H}_q(\Sigma_{2m}) \subseteq \mathcal{H}_{(1,q)}(W(\mathrm{B}_{2m})), \end{align} $$

where $\mathcal {A}(m)$ is the Hu algebra which we will define shortly. From now on, all the computation are carried out inside $\mathcal {H}^{\mathrm {B}}$ in the sense of Equation (6.1.1).

Definition 6.1.1. Let $T_{a\to b}$ and $T_{a\to b \to a}$ be the element in $\mathcal {H}^{\mathrm {B}}$ corresponding to $s_{a\to b}$ and $s_{a\to b\to a}$ (see (2.1.1)), respectively. Denote the Jucys–Murphy elements by

(6.1.2) $$ \begin{align} u_k^{\pm} = \prod_{i=0}^{k-1} (q^{i} \pm T_{i \to 0 \to i}) \in \mathcal{H}^{\mathrm{B}}. \end{align} $$

It is well defined because its factors commute pairwise. Note that an empty product is understood as 1 so $u^{\pm }_0 = 1$ . The following result is a useful variant of [Reference Dipper, James and MurphyDJM95, Lemma 4.9].

Proof. Parts (a) and (b) are immediate consequences of the definitions. For part (c), write $x = yc \in \Sigma _{2m}$ for $y \in \Sigma _i \times \Sigma _{2m-i}$ and $c \in \Sigma _{2m}$ the shortest representative in the coset $(\Sigma _i \times \Sigma _{2m-i})x$ so that $u_i^+ T_x = u_i^+ T_y T_c = T_y u_i^+ T_c$ by part (a). If c fixes 1, then $T_c$ commutes with $T_0$ , and hence,

$$ \begin{align*}u_i^+ T_xT_0 = T_y u_i^+ T_0T_c = T_y u_i^+ T_c = u_i^+ T_x\end{align*} $$

since $u_i^+ T_0 = u_i^+$ when $i \geq 1$ . We are done in this case.

If c does not fix $1$ , then $c = s_{i\to 1} z$ is a reduced expression for some $T_z$ that commutes with $T_0$ , and hence

(6.1.5) $$ \begin{align} \begin{aligned} u_i^+ T_x T_0 = u_i^+ T_y T_{i\to 1} T_z T_0 = u_i^+ T_y T_{i\to 0} T_z = T_y u_i^+ T_{i\to 0} T_z & \quad\text{by part (a)} \\ = T_y (u_{i+1}^+ T_{1\to i}^{-1} - q^i u_i^+ T_{1\to i}^{-1}) T_z & \quad\text{by part (b)} \\ = T_y u_{i+1}^+ T_{1\to i}^{-1} T_z - q^i u_i^+ T_yT_{1\to i}^{-1} T_z. & \quad\text{by part (a)} \end{aligned} \end{align} $$

As a consequence of the lemma, the definition below makes sense as one can expand $T_{w_{m,m}} u_m^-$ and then apply Equation (6.1.3) iteratively to define $h_{m}$ .

Definition 6.1.3. Let $h_m$ be the unique element in $\mathcal {H}:= \mathcal {H}_q(\Sigma _{2m})$ such that

(6.1.6) $$ \begin{align} u_m^+T_{w_{m,m}} u_m^- \in u^+_m h_m + \sum_{b>m} \mathcal{H} u_b^+ \mathcal{H}. \end{align} $$

6.2 Definition of $\mathcal {A}(m)$

With the construction of $h_m$ , one can now define the Hu algebra.

Definition 6.2.1 (Hu algebra).

Let $\mathcal {A}(m)$ be the subalgebra of $\mathcal {H}_q(\Sigma _{2m})$ generated by

(6.2.1) $$ \begin{align} T_1, \ldots, T_{m-1}, T_{m+1}, \ldots, T_{2m-1}, H_1, \end{align} $$

where $H_1= h_m^*$ , and $*$ is the unique antiautomorphism of $\mathcal {H}_q(\Sigma _{2m})$ determined by $T_i^* = T_i$ for all $1\leq i < 2m$ .

Several of the important properties that Hu proved about $\mathcal {A}(m)$ are summarized below.

6.3 A new formula for $H_{1}$

In this section, we assume that there is an invertible element $v\in K^\times $ such that $v^{2} = q$ . This is true when K is an algebraically closed field and when K is the ring of Laurent polynomials in v. The main goal of this section is to give a new and useful interpretation of $H_{1}$ .

For simplicity, we consider a renormalized standard basis $\{I_w := v^{-\ell (w)}T_w~|~ w\in \Sigma _{2m}\}$ of $\mathcal {H}_q(\Sigma _{2m})$ satisfying, for all $I_i := I_{s_i}$ ,

(6.3.1) $$ \begin{align} (I_i +v^{-1})(I_i-v)= 0, \quad \text{or} \quad I_i^2 = (v -v^{-1}) I_i +1. \end{align} $$

Denote by $\overline {\phantom {a}}: \mathcal {H}_q({\Sigma _{2m}}) \to \mathcal {H}_q({\Sigma _{2m}})$ the bar involution sending v to $v^{-1}$ and $I_w$ to $I_{w^{-1}}^{-1}$ . We use the shorthand notation

(6.3.2) $$ \begin{align} I_{a\to b}^+ := I_{a\to b}, \quad I_{a\to b}^- := \overline{I_{a\to b}} = I_{b\to a}^{-1}. \end{align} $$

Denote by $\{\pm \}^m := \{+, -\}^m$ the set of m-tuples of plus or minus signs. For $\epsilon = (\epsilon _1, \dots , \epsilon _m) \in \{\pm \}^m$ , let

(6.3.3) $$ \begin{align} I_\epsilon = I_{m\to1}^{\epsilon_1} I_{m+1\to2}^{\epsilon_2} \dots I_{2m-1\to m}^{\epsilon_m}, \end{align} $$

and let $\Sigma _d$ act on $\{\pm \}^m$ by place permutations.

Proof. (a) This is a direct consequence of the braid relations. One obtains a graphical proof by treating $I_i$ and $I_i^{-1}$ as opposite crossings of the ith and the $(i+1)$ th strings.

(b) It suffices to prove the case $a=b+1$ . Note that

(6.3.6) $$ \begin{align} I_{a\to c} I_{c\to a} I_b = I_{a\to c} I_{c\to b-1} I_b I_a I_b = I_{a\to c} I_{c\to b-1} I_a I_b I_a = I_a I_b I_a I_{b-1 \to c} I_{c\to a} = I_b I_{a\to c} I_{c\to a}, \end{align} $$

it then follows by an inductive argument that $I_b, \dots , I_c$ (and hence $I_{b\to c} I_{c\to b}$ ) commutes with $I_{a\to c} I_{c\to a}$ .

(c) The first assertion follows from Part (b) since $c_{m,i} c_{m,j} = \pi _m (I_{i\to 1}I_{1\to i} I_{j\to 1} I_{1\to j})$ . The second assertion is an immediate consequence of the first one.

Lemma 6.3.2. Let $h_{m,i}$ be the unique element in $\mathcal {H}:=\mathcal {H}_q(\Sigma _{2m})$ such that

(6.3.7) $$ \begin{align} u_m^+ T_{w_{m,i}} u_i^- \in u_m^+ h_{m,i} + \sum_{b>m} \mathcal{H} u_b^+ \mathcal{H}. \end{align} $$

Then, the elements $h_{m,i} (1\leq 1 \leq m)$ are given by the formula below recursively:

(6.3.8) $$ \begin{align} \begin{aligned} h_{m,1} &= v^m(I_{m \to 1} + I^-_{m \to 1}), \\ h_{m,i+1} &= v^{m+2i} (h_{m,i}I_{i+m \to i+1} + c_{m,i} h_{m,i} I^-_{i+m \to i+1}). \end{aligned} \end{align} $$

In particular, $h_m = h_{m,m}$ is given by

(6.3.9) $$ \begin{align} h_m = v^{m(2m-1)}\sum_{\epsilon \in \{+,-\}^m} C_\epsilon I_{m\to1}^{\epsilon_1} I_{m+1\to2}^{\epsilon_2} \dots I_{2m-1\to m}^{\epsilon_m}. \end{align} $$

Proof. The base case $i=1$ follows immediately from Lemma 6.1.2(c). For the inductive step, by definition of $u_i^-$ , we have

(6.3.10) $$ \begin{align} u_m^+ T_{w_{m,i+1}} u_{i+1}^- = q^{i} u_m^+ T_{w_{m,i+1}} u_i^- - u_m^+ T_{w_{m,i+1}} T_{i\to0\to i} u_i^-. \end{align} $$

The first term on the right-hand side of Equation (6.3.10) is

(6.3.11) $$ \begin{align} q^{i} u_m^+ T_{w_{m,i+1}} u_i^- = q^{i} u_m^+ T_{w_{m,i}} u_i^- T_{m+i\to i+1} = u_m^+ q^{i} h_{m,i} T_{m+i\to i+1} + \sum_{b>m } \mathcal{H} u_b^+ \mathcal{H}, \end{align} $$

and hence its contribution to $h_{m,i+1}$ is $ q^{i} h_{m,i} T_{i+m\to i+1} = v^{m+2i} h_{m,i} I_{i+m\to i+1}$ . The second term on the right-hand side of Equation (6.3.10) is

(6.3.12) $$ \begin{align} \begin{array}{ll} - u_m^+ T_{w_{m,i+1}} T_{i\to1} T_{0\to i} u_i^- \\ \quad = - u_m^+ T_{i+m\to1+m} T_{w_{m,i+1}} T_{0\to i} u_i^- &\text{Proposition }{6.3.1}(a) \\ \quad= - T_{m+i\to m+1} u_m^+ T_{m\to 1} (T_{m+1\to 2} \dots T_{i+m \to i+1}) T_{0\to i} u_i^- &\text{Lemma }{6.1.2}(a) \\ \quad = - T_{m+i\to m+1} u_m^+ T_{m\to 0} (T_{m+1\to 2} \dots T_{i+m \to i+1}) T_{1\to i} u_i^- \\ \quad= q^m T_{m+i\to m+1} u_m^+ T^-_{m\to 1} T_{m+1\to 2} \dots T_{i+m \to i+1} T_{1\to i} u_i^- + \sum_{b>m } \mathcal{H} u_b^+ \mathcal{H} &\text{Lemma }{6.1.2}(b) \\ \quad = q^m T_{m+i\to m+1}T_{m+1\to m+i} u_m^+ T_{w_{m,i}} T^-_{i+m \to i+1} u_i^- + \sum_{b>m } \mathcal{H} u_b^+ \mathcal{H} &\text{Proposition }{6.3.1}(a) \\ \quad = q^m c_{m,i} u_m^+ T_{w_{m,i}}u_i^- T^-_{i+m \to i+1} + \sum_{b>m } \mathcal{H} u_b^+ \mathcal{H} &\text{Lemma }{6.1.2}(a) \\ \quad= q^m c_{m,i} u_m^+ h_{m,i} T^-_{i+m \to i+1} + \sum_{b>m } \mathcal{H} u_b^+ \mathcal{H}, \end{array} \end{align} $$

and hence its contribution to $h_{m,i+1}$ is $v^{m+2i} c_{m,i} h_{m,i} I^-_{i+m \to i+1}$ , and we are done.

Theorem 6.3.3. We have an explicit formula for $H_1 = H_1(m) :=h_m^*$ as follows:

(6.3.13) $$ \begin{align} H_1 = v^{m(2m-1)}\sum_{\epsilon \in \{+,-\}^m} I_\epsilon C_\epsilon, \quad \text{where} \quad I_\epsilon = I_{m\to 2m-1}^{\epsilon_1} \dots I_{1\to m}^{\epsilon_m}, \quad C_\epsilon = \prod_{i;\epsilon_i=-}c_{m,m-i}. \end{align} $$

Example 6.3.4. Consider $H_1 \in \mathcal {A}(2) \subseteq \mathcal {H}_q(\Sigma _4)$ . We use abbreviations such as $\overline {T}_2 T_1 \overline {T}_3 \overline {T}_2 := T_{\overline {2}.1.\overline {3.2}}$ . Then $h_{2} = v^6 (I_{2.1.3.2} + I_{2.1.\overline {3.2}} + I_3^2(I_{\overline {2.1}.3.2} + I_{\overline {2.1}.\overline {3.2}}){)}$ , or

(6.3.14) $$ \begin{align} H_1 = v^6( I_{2.3.1.2} + I_{2.3.\overline{1.2}} + (I_{\overline{2.3}.1.2} + I_{\overline{2.3}.\overline{1.2}}) I_3^2{)}. \end{align} $$

While $I_{2.1.3.2}^2$ consists of summands involving $I_2$ in terms of the standard basis, its alternative, $H_1$ , squares into the parabolic subalgebra $\mathcal {H}_q(\Sigma _2 \times \Sigma _2) = \langle I_1, I_3\rangle $ . To be precise, we obtained an explicit formula for the central element $z_{2,2} = H_1^2$ (see Proposition 6.2.2(c)):

(6.3.15) $$ \begin{align} z_{2,2} =\ &v^6(q-1)^2(q^4+2q^3-2q^2+2q+1) I_{31} \nonumber\\ &+v^7(q-1)(q^4+4q^3-2q^2+4q+1)(I_1+I_3) \\ &+2v^8(q^4+4q^3-2q^2+4q+1).\nonumber \end{align} $$

We currently have no explanations for these coefficients appearing in $z_{m,m}$ .

6.4 Bar-invariant bases

In this section, we assume $K = \mathbb {Z}[v^{\pm 1}]$ . The element $H_1$ is defined via solving an equality (see Definition 6.1.3) that involves Jucys–Murphy elements in the type B Hecke algebra of unequal parameters. There seems to be no a priori reason for $H_1$ to admit an explicit formula in Hecke algebra of Type A. In this section, we show miraculously that $H_1$ , after being multiplied by an element arising from the longest element, leads to a highly nontrivial bar-invariant element $b_1$ and thus to a bar-invariant basis for the Hu algebra. The element $b_1$ seems to afford a positive expansion in terms of the dual canonical basis of the ambient Hecke algebra and in turn affords a potential theory of geometric realization and/or categorification for such wreath product $\Sigma _m \wr \Sigma _d$ that are not Coxeter groups in general.

Proof. Part (a) follows from the fact that $s_1 s_{2\to 1} \dots s_{m-1\to 1} = w_\circ (m)$ is a reduced expression. Part (b) follows from the fact that $w_\circ (m) s_i = s_{m+1-i} w_\circ (m)$ for all i. For part (c), we prove it by an induction on m. The base case $m=2$ follows since $w_\circ (2) = s_1$ and $C= I_3^2$ as computed in Example 6.3.4. For $m \geq 3$ , we have

(6.4.1) $$ \begin{align} \begin{array}{rll} C &= c_{m,1} \dots c_{m,m-1} = \pi_m(I_1^2 I_{2.1.1.2} \dots I_{m-1\to1} I_{1\to m-1}) \\ &= \pi_m(I_{w_\circ(m-1)}^2 I_{m-1\to1} I_{1\to m-1}) &\text{via the induction hypothesis} \\ &= \pi_m (I_{w_\circ(m-1)} I_{w_\circ(m)} I_{1\to m-1}) &\text{by part (a)} \\ &= \pi_m (I_{w_\circ(m-1)} I_{m-1\to 1}I_{w_\circ(m)} ) &\text{by part (b)} \\ &= \gamma^2. &\text{by part (a)} \end{array} \end{align} $$

The extra generator $H_1$ behaves well with respect to the bar involution in the following way:

Proposition 6.4.2. The following element is bar-invariant:

(6.4.2) $$ \begin{align} b_1:= v^{-m(2m-1)} H_1 \overline{\gamma} = \sum_{\epsilon \in \{\pm\}^m} I_\epsilon C_\epsilon \overline{\gamma}. \end{align} $$

Proof. Note that for any $\epsilon \in \{\pm \}^m$ ,

(6.4.3) $$ \begin{align} \overline{I}_\epsilon = \overline{I^{\epsilon_1}_{m\to 2m-1}} \dots \overline{I^{\epsilon_m}_{1\to m}} = I^{-\epsilon_1}_{m\to 2m-1} \dots I^{-\epsilon_m}_{1\to m} = I_{\overline{\epsilon}}, \end{align} $$

where $\overline {\epsilon } := -\epsilon \in \{\pm \}^m$ . Recall that $H_1 = v^{m(2m-1)}\sum _\epsilon I_\epsilon C_\epsilon $ , so $b_1 = \sum _\epsilon I_\epsilon C_\epsilon \overline {\gamma },$ and hence

(6.4.4) $$ \begin{align} \overline{b}_1 = \sum_\epsilon \overline{I}_\epsilon \overline{C}_\epsilon \gamma = \sum_{\overline{\epsilon}} I_{\overline{\epsilon}} \overline{C}_{\epsilon} \gamma = \sum_{\epsilon} I_{\epsilon} \overline{C}_{\overline{\epsilon}} \gamma. \end{align} $$

Therefore, $b_1$ is bar-invariant as long as

(6.4.5) $$ \begin{align} C_{\epsilon} \overline{\gamma} = \overline{C}_{\overline{\epsilon}}\gamma \quad \text{ for all }\epsilon \in \{\pm\}^m. \end{align} $$

Thanks to Proposition 6.4.1(c), one has $\gamma ^2 = C = C_{\overline {\epsilon }} C_{\epsilon }$ for any $\epsilon $ , and hence Equation (6.4.5) follows from a right multiplication by $\overline {\gamma }$ and a left multiplication by $\overline {C}_{\overline {\epsilon }}$ .

With the help of this distinguished element $b_1$ , we are now able to construct a bar-invariant basis $\{b_x~|~x\in \Sigma _m\wr \Sigma _2\}$ of the Hu algebra that satisfies a unitriangular relation with respect to a standard basis $\{I_x ~|~ x\in \Sigma _m \wr \Sigma _2\}$ . See Remark 6.4.5 for a discussion on the positivity of $\{b_x\}_x$ in comparison with the canonical bases.

Theorem 6.4.4. Let $\{b_w ~|~ \Sigma _m \times \Sigma _m\}$ be the canonical basis for $\mathcal {H}_q(\Sigma _m\times \Sigma _m)$ in the sense that $b_w \in I_w + \sum _{y<w} v\mathbb {Z}[v] I_y$ , and let $b_{wt_1} := b_w b_1$ for $w\in \Sigma _m \times \Sigma _m$ . Then $\{b_x ~|~ x\in \Sigma _m \wr \Sigma _2\}$ forms a bar-invariant basis of $\mathcal {A}(m)$ satisfying

(6.4.6) $$ \begin{align} b_x \in I_x + \sum_{y < x} v\mathbb{N}[v] I_y, \end{align} $$

with respect to the Bruhat order inherited from $\Sigma _{2m}$ .

Remark 6.4.5. While the canonical basis of $\mathcal {H}_q(W)$ of any Weyl group W admits positive structural constants, that is, $b_x b_y \in \sum _z \mathbb {N}[v^{\pm 1}] b_z$ ; our bar-invariant basis does not have such positivity because $b_1^2$ has mixed signs already.

However, $b_1$ seems to admit a negative expansion in the canonical basis, equivalently, a positive expansion in the dual canonical basis $\{c_w ~|~ w\in \Sigma _{2m}\}$ , that is,

(6.4.7) $$ \begin{align} b_1 \in \sum_{z \in \Sigma_{2m}} \mathbb{N}[(-v^{-1})^{\pm1}] b_z = \sum_{z \in \Sigma_{2m}} \mathbb{N}[v^{\pm1}] c_z, \end{align} $$

where the dual basis element $c_w$ is characterized by $\overline {c}_w = c_w \in I_w + \sum _{y<w} (-v^{-1})\mathbb {N}[-v^{-1}] I_y$ , and it satisfies that $c_x c_y \in \sum _z \mathbb {N}[-v^{\pm 1}] c_z$ .

6.5

We present a concrete calculation of $b_{1}$ in a specific example below.

Example 6.5.1. For $H_1 = H_1(3)$ , write $I_{\epsilon _1 \epsilon _2}:= I_{(\epsilon _1,\epsilon _2,+)} + I_{(\epsilon _1,\epsilon _2,-)}$ for short. Then $H_1 = v^{15}(I_{++} + I_{+-}I_{4.4} + I_{-+}I_{5.4.4.5} + I_{--}I_{4.4.5.4.4.5})$ , $\gamma = I_{4.5.4}$ , and hence

(6.5.1) $$ \begin{align} \begin{aligned} b_1 &= (I_{++} + I_{+-}I_{4.4} + I_{-+}I_{5.4.4.5} + I_{--}I_{4.4.5.4.4.5})I_{\overline{4.5.4}} \\ &= (I_{++}I_{\overline{4.5.4}} + I_{+-}I_{4.\overline{5.4}} + I_{-+}I_{\overline{4}.5.4} + I_{--}I_{4.5.4}). \end{aligned} \end{align} $$

The element $b_1$ is indeed bar-invariant since

(6.5.2) $$ \begin{align} \begin{aligned} \overline{b}_1 &= (I_{--} + I_{-+}I_{\overline{4.4}} + I_{+-}I_{\overline{5.4.4.5}} + I_{++}I_{\overline{4.4.5.4.4.5}}) I_{4.5.4} \\ &= (I_{--}I_{4.5.4} + I_{-+}I_{\overline{4}.5.4} + I_{+-}I_{4.\overline{5.4}} + I_{++}I_{\overline{4.5.4}}) = b_1. \end{aligned} \end{align} $$

Examples for a positive expansion of $b_1=b_1(m)$ in the dual canonical basis are shown below: for $m=1$ ,

(6.5.3) $$ \begin{align} b_1(1) = 2b_{s_1} + (-v-v^{-1}) = 2c_{s_1} + (v+v^{-1}). \end{align} $$

For $m=2$ , we abbreviate $c_{s_{i}s_{j}\dots s_{k}}$ by $c_{i.j.\dots .k}$ , and obtain:

(6.5.4) $$ \begin{align} b_1(2)\,{=}\,&4c_{2.3.1.2.3} + 2(v+v^{-1})(c_{2.3.1.2}+c_{2.1.2.3}+c_{3.1.2.3})\nonumber \\ &{+}\, 4(c_{2.1.2}+c_{3.1.2}) +2(v^2+v^{-2})c_{1.2.3} +2(v+v^{-1})(c_{2.1}+c_{1.2}+c_{3.2}+c_{2.3}) \\ &{+}\,(v^2+2+v^{-2})(c_1+2c_2+c_3) +(v^3+v+v^{-1}+v^{-3}). \nonumber \end{align} $$

6.6 Generalized Hu algebra

With the explicit formula above, we are in a position to define a Hecke subalgebra $\mathcal {H}_{m\wr d}$ of $\mathcal {H}_q(\Sigma _{md})$ for the wreath product $\Sigma _m \wr \Sigma _d$ . Note that Hu’s Hecke subalgebra is a special case $\mathcal {A}(m) = \mathcal {H}_{m \wr 2}$ .

Definition 6.6.1. Let $\mathcal {H}_{m\wr d}$ be the subalgebra of $\mathcal {H}_q(\Sigma _{md})$ generated by

$$\begin{align*}T_i, ( i \in [md]\setminus m\mathbb{Z}), \quad H_j, (1\leq j \leq d-1), \end{align*}$$

where $H_{j+1} = \pi _{jm}(h^*_m)$ (see Equation (6.3.4)). The algebra $\mathcal {H}_{m \wr d}$ will be called the generalized Hu algebra with $\mathcal {H}_{m \wr 2} = \mathcal {A}(m)$ .

The braid relations fail for the elements $H_i$ ’s. For example, in the special case when $m=1$ , the $H_i$ ’s satisfy the modified braid relations

(6.6.1) $$ \begin{align} H_i H_{i+1} H_i - H_{i+1} H_iH_{i+1} = (q-1)^2 (H_{i+1} - H_i), \end{align} $$

which appeared in the nonstandard presentation of the type A Hecke algebra (cf. [Reference WangWa07]). In other words, Wang’s nonstandard presentation of $\mathcal {H}_q(\Sigma _d)$ is just the ‘standard’ presentation of the generalized Hu algebra $\mathcal {H}_{1\wr d}$ . This suggests that there could be a general theory for covering algebras for $\mathcal {H}_{m\wr d}$ .

7.1

In this section, we investigate the connection between various properties that are referred as the ‘Schur–Weyl duality’, including the double centralizer property, the existing of a splitting, and the existence of a certain idempotent. A systematical approach is provided to obtain the Schur–Weyl duality for the quantum wreath product $A = B \wr \mathcal {H}(d)$ arising from a Schur–Weyl duality for the base algebra B. As a consequence, the Schur–Weyl duality for the Hu algebra $\mathcal {A}(m) = \mathcal {H}_q(\Sigma _m) \wr \mathcal {H}(2)$ is established for the first time. It should be noted that our approach is quite encompassing and allows us to recover the Schur–Weyl duality for other Hecke-like algebras appearing in Section 4.

7.2

Various constructions have appeared in the literature that have involved endomorphism algebras and the double centralizer property (cf. [Reference Cline, Parshall and ScottCPS96, Chapter 1] [Reference Doty, Erdmann and NakanoDEN04, Sections 2-3]). In familiar cases like commuting actions of the general linear group and the symmetric group, the ‘Schur functor’ is constructed via an explicit idempotent. For more general settings, idempotents in algebras are not easy to locate. For the purposes of this paper, we present an alternative approach via splittings to prove the existence of an idempotent $e\in S$ (where S is an endomorphism algebra) that can be used to construct a functor from $\text {Mod}(S)$ to $\text {Mod}(eSe)$ .

For now, assume that A is an arbitrary associative K-algebra, and T is a right A-module with $S=\operatorname {End}_{A}(T)$ . Then T is a left S-module (under composition), and T is an S-A-bimodule. Suppose one has a splitting, as right A-modules,

(7.2.1) $$ \begin{align} T\cong A\oplus Q \quad (\text{for some right }A\text{-module }Q). \end{align} $$

There exists a contravariant functor ${\mathcal G}(-):=\text {Hom}_{A}(-,T)$ whose image is a left S-module. Under this functor, ${\mathcal G}(A)=\text {Hom}_{A}(A,T)\cong T$ is a direct summand of S as a left S-module by using the splitting. Consequently, T is a projective left S-module.

Under the condition (7.2.1), there exists an idempotent $e\in S$ such that $T\cong Se$ . Moreover,

$$ \begin{align*}\operatorname{End}_{S}(T)\cong \operatorname{End}_{S}(Se)\cong eSe.\end{align*} $$

One can define a Schur functor ${\mathcal F}:\text {Mod}(S) \rightarrow \text {Mod}(eSe)$ via ${\mathcal F}(M)=eM$ . In this setting, the theory of Schur functors and their inverses has been developed in [Reference Doty, Erdmann and NakanoDEN04].

7.3

The question remains when one can identify A with $eSe$ . When this holds the aforementioned Schur functor will conveniently take objects in $\text {Mod}(S)$ to $\text {Mod}(A)$ . When A is a self-injective algebra the following result is a corollary of a theorem proved by König, Slungard and Xi [Reference König, Slungard and XiKSX01]. The original statement of the corollary first appeared in [Reference Curtis and ReinerCR62, 59.6].

Then $\operatorname {End}_{S}(T)\cong A$ .

Condition (a) of the theorem will be satisfied when A is a quantum wreath product as long as the base algebra B is a symmetric algebra and the assumptions in Proposition 3.4.1(b) hold. For examples of quantum wreath products, to ensure the existence of a natural Schur functor, one needs to establish a splitting of T as in (7.2.1). This will ensure that condition (b) of the theorem is satisfied.

7.4 A splitting lemma, I

For applications in Schur–Weyl duality, we make the following assumptions:

(7.4.1) $$ \begin{align} \sigma = \text{flip}, \quad\rho = 0, \quad\mathrm{and\ Equations}\ ({3.2.1}) \text{--} ({3.2.9}) \text{ hold}. \end{align} $$

Within Section 7.4, we also assume that

(7.4.2) $$ \begin{align} r^2 = \epsilon(S)r + \epsilon(R) \quad \text{for some} \quad r\in K. \end{align} $$

Let $V_B$ be the vector space over K with basis $\{v_i ~|~ i \in I\}$ , where I is a (possibly infinite) poset that is totally ordered. Assume that $V_B$ is a right B-module, and hence $V_B^{\otimes d}$ is a right module over $B^{\otimes d}$ via the factorwise action. For $\mu = (\mu _j)_j \in I^d$ , set $v_\mu = v_{\mu _1} \otimes \ldots \otimes v_{\mu _d} \in V_B^{\otimes d}$ . Assume that there is $\mu \in I^d$ such that $\mu _1 < \dots < \mu _d$ , and let $W = W(\mu )$ be the subspace of $V_B^{\otimes d}$ spanned by elements of the form $v_{\mu \cdot g}$ , where $\Sigma _d$ acts on $I^d$ by place permutations. Let the Hecke-like generators $H_i$ ’s in $A = B \wr \mathcal {H}(d)$ act on the $B^{\otimes d}$ -submodule W from the right by

(7.4.3) $$ \begin{align} v_\mu. H_i = \begin{cases} v_{\mu \cdot s_i}&\text{if } \mu_i< \mu_{i+1}; \\ v_{\mu \cdot s_i}.{R_i} + v_\mu.{S_i}&\text{if } \mu_i> \mu_{i+1}. \end{cases} \end{align} $$

Thanks to the assumption (7.4.2) and that I is totally ordered, the action (7.4.3) on W extends to the following action on $V_B^{\otimes d}$ , as long as Equation (7.4.3) is well defined:

(7.4.4) $$ \begin{align} v_\mu. H_i = \begin{cases} v_{\mu \cdot s_i}&\text{if } \mu_i< \mu_{i+1}; \\ r v_\mu &\text{if } \mu_i = \mu_{i+1}; \\ v_{\mu \cdot s_i}.{R_i} + v_\mu.{S_i}&\text{if } \mu_i> \mu_{i+1}. \end{cases} \end{align} $$

We now show that Equation (7.4.3) is well defined in two separate cases. The first case is when B acts on $V_B$ (and hence on the tensor products of $V_B$ ) by a counit $\epsilon :B \to K$ . The second case is the degenerate case when A is just a twisted tensor product of $B^{\otimes d}$ by a group algebra $K\Sigma _d$ .

Proof. For (a), the verification of the quadratic relations reduces to the rank two case: for $i < j$ , $(v_{(i,j)} H_1) H_1 = v_{(j,i)} H_1 = \epsilon (S) v_{(j,i)} + \epsilon (R) v_{(i,j)} = v_{(i,j)} (H_1^2)$ . The other case $i> j$ can be checked similarly. The verification of the braid relations reduces to the rank three case. We only present the most complicated case for $i < j < k$ :

(7.4.5) $$ \begin{align} \begin{aligned} &v_{(k,j,i)}H_1 H_2 H_1 = (\epsilon(S) v_{(k,j,i)} + \epsilon(R) v_{(j,k,i)})H_2H_1 \\ &\quad =(\epsilon(S^2) v_{(k,j,i)} + \epsilon(SR)(v_{(j,k,i)} + v_{(k,i,j)}) + \epsilon(R^2)v_{(j,i,k)})H_1 \\ &\quad =\epsilon(S^3+SR) v_{(k,j,i)} + \epsilon(S^2R)(v_{(j,k,i)} + v_{(k,i,j)}) + \epsilon(SR^2)(v_{(i,k,j)} + v_{(j,i,k)}) + \epsilon(R^3)v_{(i,j,k)} \\ &\quad =(\epsilon(S^2) v_{(k,j,i)} + \epsilon(SR)(v_{(j,k,i)} + v_{(k,i,j)}) + \epsilon(R^2)v_{(i,k,j)})H_2 = v_{(k,j,i)}H_2 H_1 H_2. \end{aligned} \end{align} $$

When $i=j$ , the braid relations hold due to a case-by-case analysis similar to the ones for $i\neq j$ . The verification of quadratic relations reduces to the rank two case:

$$ \begin{align*}(v_{(i,i)} H_1) H_1 = r^2 v_{(i, i)} = \epsilon(S) v_{(i,i)} + \epsilon(R) v_{(i,i)} = v_{(i,i)} (H_1^2).\end{align*} $$

The wreath relations hold since $\rho = 0$ and the fact that both $a\otimes b, b\otimes a \in B\otimes B$ act on $V_B^{\otimes 2}$ by the same scalar multiple $\epsilon (a)\epsilon (b)$ .

For (b), the A-action on W is given explicitly by $v_\mu. H_i = v_{\mu \cdot s_i}$ . The braid and quadratic relations are immediate, while the wreath relations follow from the rank two verification:

$$ \begin{align*}v_{(i,j)} \sigma(b) H_1 = (\sum_{k} v_i b^{\prime\prime}_k \otimes v_j b^{\prime}_k) H_1 = v_{(j,i)} b = v_{(i,j)} H_1 b,\end{align*} $$

where $b = \sum _k b^{\prime }_k \otimes b^{\prime \prime }_k \in B\otimes B$ .

Schur–Weyl duality in the literature often assumes a condition $n \geq d$ for the double centralizer property to hold. In the following lemma, we demonstrate that the condition $n\geq d$ is essentially a shadow of the assumption that $V_B$ contains d distinct direct summands which are isomorphic to B as B-modules. To be precise, one needs to assume additionally the subspace W obtained from these direct summands admits an A-action given by Equation (7.4.3) and right multiplications by $B^{\otimes d}$ (e.g., the two cases in Proposition 7.4.1). In such a case, $W \cong A$ as A-modules, and hence the desired splitting is obtained when the A-action on W extends to $V_B^{\otimes d}$ .

Lemma 7.4.2. Assume that Equation (7.4.1) holds, and there are right B-module isomorphisms $\phi _j: U_j \to B, v_{\mu _j} b \mapsto b$ such that $U_j$ ’s are distinct direct summands of $V_B$ .

If Equation (7.4.3) extends to an A-action on $W=W(\mu )$ , then $W \simeq A$ as right A-modules. If Equation (7.4.3) extends further to an A-action on $V_B^{\otimes d}$ , then $V_B^{\otimes d}$ has the following splitting, as right A-modules,

$$\begin{align*}V_B^{\otimes d} \simeq A \oplus Q. \end{align*}$$

Proof. We may assume that ${\mu _1} < {\mu _2} < \dots < {\mu _d}$ . It suffices to show that the assignment

(7.4.6) $$ \begin{align} \psi:v_\mu.a \mapsto a \quad (\text{for all }a \in A) \end{align} $$

defines an A-module isomorphism $W \to A$ . We show first that $\psi $ is well defined. Suppose that $v_\mu H_w b = v_\mu H_{w'}b'$ for some $w, w'\in \Sigma _d, b,b' \in B^{\otimes d}$ . By Equation (7.4.3), $v_\mu H_w = v_{\mu \cdot w} \in U_{w(1)} \otimes \cdots U_{w(d)}$ so is $v_\mu H_w b$ because $U_j$ ’s are summands. Since $U_j$ ’s are all distinct, w must agree with $w'$ . Therefore, $b = (\phi _{w(1)} \otimes \dots \otimes \phi _{w(d)})(v_{\mu \cdot w} b) =(\phi _{w(1)} \otimes \dots \otimes \phi _{w(d)})(v_{\mu \cdot w} b') = b'$ .

Next, it is easy to verify that $\psi $ is an epimorphism. For injectivity, thanks to the basis theorem, an arbitrary element in A is of the form $a= \sum _{w\in \Sigma _d} H_w (b_1^{(w)} \otimes \dots \otimes b_d^{(w)})$ for some $b^{(w)}_i \in B$ . Since ${\mu _1} < {\mu _2} < \dots < {\mu _d}$ , by Equation (7.4.3), $H_w$ acts on $v_{({\mu _1}, {\mu _2}, \dots , {\mu _d})}$ by place permutation.

Suppose that $v_{({\mu _1}, {\mu _2}, \dots , {\mu _d})}.\sum _{w\in \Sigma _d} H_w (b_1^{(w)} \otimes \dots \otimes b_d^{(w)}) = 0$ . Each $v_{({\mu _1}, {\mu _2}, \dots , {\mu _d})} H_w (b_1^{(w)} \otimes \dots \otimes b_d^{(w)})$ lies in a different summand $U_{w(\mu _1)} \otimes \dots \otimes U_{w(\mu _d)} \subseteq V_B^{\otimes d}$ , and hence $b_1^{(w)} \otimes \dots \otimes b_d^{(w)}$ must be zero for all w.

Finally, a double centralizer property is obtained in the context of Theorem 7.3.1, where the S-A-bimodule T is $V_B^{\otimes d}$ .

7.5 Examples

7.5.1 Hecke algebras of type A

We retain the setup as in Section 4.2.2 and then set $m=1$ . Therefore, $B=K$ and $A=B \wr \mathcal {H}(d)$ is just the Hecke algebra of type A. Note that in this degenerate case, X is identified with $q_1 \in K^\times $ and $\rho = 0$ . Let $V_B=V(n)$ be the K-space spanned by $v_i$ where $i \in I := \{1, \dots , n\}$ with the usual total order. Thus, $V_B^{\otimes d}$ admits an A-action by Proposition 7.4.1(a). To be precise, for $1\leq i \leq d-1$ ,

(7.5.1) $$ \begin{align} v_\mu. H_i = \begin{cases} v_{\mu \cdot s_i}&\text{if } \mu_i< \mu_{i+1}; \\ q v_{\mu \cdot s_i} &\text{if } \mu_i= \mu_{i+1}; \\ qv_{\mu \cdot s_i} + (q-1)v_\mu&\text{if } \mu_i> \mu_{i+1}. \end{cases} \end{align} $$

The assumptions in Lemma 7.4.2 hold if and only if $V(n)$ contains a d-dimensional subspace, that is, $n\geq d$ , which is the well-known condition to afford the quantum Schur–Weyl duality [Reference JimboJ86, Reference Dipper and JamesDJ89]. Moreover, $S^A(V_B^{\otimes d})$ coincides with the type A q-Schur algebra $S_q(n,d) := \operatorname {End}_{\mathcal {H}_q(\Sigma _d)}(V(n)^{\otimes d})$ .

Corollary 7.5.1. If $n\geq d$ , then the Schur duality holds between the Hecke algebra $\mathcal {H}_q(\Sigma _{d})$ and its corresponding q-Schur algebra $S_q(n,d)$ . Moreover, a Schur functor exists.

7.6 A splitting lemma, II

While in Section 7.4 we demonstrate an application of Theorem 7.3.1 via a construction of the splitting over the quantum wreath product $B \wr \mathcal {H}(d)$ from a splitting over the base algebra B, in this section we provide a new application of Theorem 7.3.1 via a different construction of the splitting. To be precise, given an algebra $A'$ , we can obtain the desired splitting over $A'$ as long as we have a splitting over a larger algebra A which contains $A'$ as a self-injective subalgebra.

7.6.1 Hu algebras

Following Section 4.1, let $B = \mathcal {H}_q(\Sigma _m)$ be generated by $T_1, \dots , T_{m-1}$ , and let $\mathcal {A}(m) = B\wr \mathcal {H}(2)$ the Hu algebra generated by $B^{\otimes 2}$ and an Hecke-like generator H. Next, we will apply Lemma 7.6.1 to the case $A' := \mathcal {A}(m) \subseteq A := \mathcal {H}_q(\Sigma _{2m})$ .

Let $V(n)$ be the K-span of $v_1, \dots , v_n$ , and let $T = T(n, 2 m) := V(n)^{\otimes 2m}$ . The Hecke algebra $\mathcal {H}_q(\Sigma _{2m})$ (and hence the Hu algebra $\mathcal {A}(m)$ ) acts on T by Equation (7.5.1). Therefore, we obtain the following diagram regarding the relationship between the q-Schur algebra $S_q(n,2m)$ and the wreath Schur algebra $S^{\mathcal {A}(m)}(T)$ .

(7.6.1)

Corollary 7.6.2. If $n\geq 2m$ , then the Schur duality holds between the Hu algebra $\mathcal {A}(m)$ and its corresponding Schur algebra $S^{\mathcal {A}(m)}(T(n,2m))$ . Moreover, a Schur functor exists.

Proof. Recall that $A := \mathcal {H}_q(\Sigma _{2m}) \supseteq A' := \mathcal {A}(m) \cong B \wr \mathcal {H}(2)$ , where $B = \mathcal {H}_q(\Sigma _m)$ is symmetric, $\sigma $ is the flip map, $\rho = 0$ and $R = z_m$ is invertible. Thus, the Hu algebra is symmetric thanks to Lemma 3.4.1(b), and hence $A'$ is a self-injective subalgebra of A.

When $n \geq 2 m$ , one has a splitting $T \cong A \oplus Q$ as A-modules. Then, by Lemma 7.6.1, one obtains another splitting $T \cong \mathcal {A}(m) \oplus Q'$ as $\mathcal {A}(m)$ -modules. Therefore, we can apply Theorem 7.3.1 and then obtain the desired Schur duality. The existence of a Schur functor follows from the discussion in Section 7.2.

7.7 Related Schur–Weyl dualities

In this section, we consider Schur–Weyl dualities in which the corresponding base algebras are infinite-dimensional. In this case, the notion of symmetric algebra is not defined. In turns out that we can still construct a natural A-action to afford the splitting (7.2.1), and in turn state the most natural double centralizer property which awaits for a uniform proof.

We also include a proof of Schur duality for the Ariki–Koike algebras using Theorem 7.3.1. Note that our splitting lemma does not apply here.

7.7.1 Affine Hecke algebras

Following Section 4.2.1, we have $B= K[X^{\pm 1}]$ and $A = B\wr \mathcal {H}(d)$ is isomorphic to the extended affine Hecke algebra $\mathcal {H}_q^{\mathrm {ext}}(\Sigma _d)$ of type A. Let $n\in \mathbb {N}$ , and let $V_B = \widehat {V}(n)$ be the space spanned by $\{v_i ~|~ i\in \mathbb {Z}\}$ on which $X^{\pm 1}$ acts by

(7.7.1) $$ \begin{align} v_i.X = v_{i+n}, \quad v_i.X^{-1} = v_{i-n}. \end{align} $$

If $n \geq d$ , then the assumptions in Lemma 7.4.2 hold. In this case, $U_j := v_j.B (1\leq j \leq d)$ are indeed distinct summands.

One can extend the right A-action to the entire tensor space $T = V_B^{\otimes d}$ via Equation (7.5.1). Then $S^A(T)$ coincides with the affine q-Schur algebra of type A. The corresponding Schur–Weyl duality is proved in [Reference Chari and PressleyCP94, Reference GreenGr99] as below:

Proposition 7.7.1 [Reference Chari and PressleyCP94, Reference GreenGr99].

If $n\geq d$ , then the Schur duality holds between the affine Hecke algebra $\mathcal {H}^{\mathrm {ext}}_q(\Sigma _{d})$ and its corresponding affine q-Schur algebra $S^{\mathrm {aff}}_q(n,d) := \operatorname {End}_{\mathcal {H}^{\mathrm {ext}}_q(\Sigma _d)}(\widehat {V}(n)^{\otimes d})$ .

7.7.2 Ariki–Koike algebras

Consider $F = \frac {{K[X]}}{\prod _{i=1}^m (X- q_i)}$ , where $q_i$ ’s are parameters as $v_i$ in Section 4.2.2. Let $n\in \mathbb {N}$ , and let $V_F=V(m,n)$ be the F-module with basis $\{v_1, \dots , v_{mn}\}$ , on which the F-action is given by, for $1\leq j \leq m, 0 \leq k \leq n-1$ ,

(7.7.2) $$ \begin{align} v_{km+j}. X = \begin{cases} v_{km+j+1} &\text{if } j < m; \\ \sum_{i=1}^{m} (-1)^{i-1} e_i(q_1, q_2, \dots, q_m) v_{km+m-i}&\text{if } j=m, \end{cases} \end{align} $$

where $e_i$ ’s are the elementary symmetric functions that appear in rewriting $\prod _{i=1}^m (X-q_i) =0$ into $X^m = \sum _{i=1}^{m} (-1)^{i-1} e_i(q_1, \dots , q_m) X^{m-i}$ .

Next, we define an A-module T such that $T = V_F^{\otimes d}$ as a vector space. Each $H_i (1\leq i \leq d-1)$ acts by Equation (7.5.1), and each generator $\overline {X^{(i)}} \in A$ acts on the right by $H_{i-1}\dots H_1 X^{(1)} H_1 \dots H_{i-1}$ , where $X^{(1)}$ acts on the first tensor factor by Equation (7.7.2).

If $n \geq d$ , then the splitting as in Equaton (7.2.1) exists since the span of $\{v_\mu ~|~ 1\leq \mu _i \leq md\}$ is a direct summand of T that is isomorphic to A as a right A-module. It is known that A is symmetric (and hence self-injective) by [Reference Malle and MathasMM98, Theorem 5.1], and thus, by Theorem 7.3.1, we cover the results on Schur duality in [Reference GreenGr97, Reference Bao and WangBW18, Reference Bao, Wang and WatanabeBWW18]. Moreover, when $m> 2$ , $S^A(T)$ is a new Schur algebra for the Ariki–Koike algebra that has the double centralizer property. See [Reference Sakamoto and ShojiSS99, Reference James and MathasJM00] for attempts to obtain such a Schur duality.

Corollary 7.7.2. If $n\geq d$ , then the Schur duality holds between the Ariki–Koike algebra $\mathcal {H}_{q, q_1, \dots , q_m}(C_m \wr \Sigma _{d})$ and its corresponding Schur algebra $S^A(V(m,n)^{\otimes d})$ .

7.7.3 Affine Hecke algebras of type C

Following Section 4.2.3, let $F = \mathcal {H}_q(\Sigma ^{\mathrm {aff}}_2)$ be generated by $X,Y$ modulo the quadratic relations $X^2 = (\xi ^{-1} - \xi )X + 1$ and $Y^2 = (\eta ^{-1} -\eta )Y +1$ for parameters $\xi , \eta \in K^{\times }$ . Let $V_F = \widetilde {V}(n)$ be the space spanned by $\{v_i ~|~ i\in \mathbb {Z}\}$ , on which F acts by

(7.7.3) $$ \begin{align} v_i. X = \begin{cases} v_{-i} &\text{if } j> 0; \\ v_{-i} + (\xi^{-1} - \xi) v_i &\text{if } j <0, \end{cases} \quad v_i. Y = \begin{cases} v_{2(n+1)-i} &\text{if } j < n+1; \\ v_{2(n+1)-i} + (\eta^{-1} - \eta) v_i &\text{if } j> n+1. \end{cases} \end{align} $$

For the right A-module structure on $T = {V}_F^{\otimes d}$ , $H_i$ ’s act by Equation (7.5.1), $\overline {X^{(i)}}, \overline {Y^{(d-i+1)}} \in A$ act on the right by $H_{i-1}\dots H_1 X^{(1)} H_1 \dots H_{i-1}$ and $H_{d-i} \dots H_{d-1}Y^{(d)}H_{d-1} \dots H_{d-i}$ , respectively, where $X^{(1)}$ and $Y^{(d)}$ act on the first and last tensor factors, respectively, by Equation (7.7.3). Then $S^A(T)$ coincides with the affine q-Schur algebra of type C. The corresponding Schur–Weyl duality, after a renormalization, is proved in [Reference Fan, Lai, Li, Luo, Wang and WatanabeFL³W²20] as below:

Proposition 7.7.3 [Reference Fan, Lai, Li, Luo, Wang and WatanabeFL³W²20].

If $n\geq d$ , then the Schur duality holds between the affine Hecke algebra $\mathcal {H}_{q,\xi ,\eta }(C^{\mathrm {aff}}_{d})$ and its corresponding affine q-Schur algebra $\widehat {S}^{\mathfrak {c}}_{q,\xi ,\eta }(n,d) := \operatorname {End}_{\mathcal {H}_{q,\xi ,\eta }(C^{\mathrm {aff}}_{d})}(\widetilde {V}(n)^{\otimes d})$ .

7.8 Connections to quantum groups

In this article, we only discuss a Schur duality between a quantum wreath product and its corresponding Schur algebra; while in literature, there is usually a third object that surjects onto the Schur algebra. For example, it is the Drinfeld–Jimbo quantum group that surjects onto the q-Schur algebras of type A.

A standard procedure due to Beilinson–Lusztig–MacPherson can be used to produce these higher level algebras from the given families of Schur algebras. See [Reference Beilinson, Lusztig and MacPhersonBLM90] for constructing the quantum group $U_q(\mathfrak {gl_n})$ from $\mathcal {H}_q(\Sigma _d)$ , and [Reference Bao, Kujawa, Li and WangBKLW18, Reference Lai and LuoLL21] for a realization of equal or multiparameter $\imath $ quantum groups $\imath U(\mathfrak {gl}_n)$ from Hecke algebra of type B/C. Furthermore, [Reference Fan, Lai, Li, Luo and WangFL³W23] explains how to obtain the affine $\imath $ quantum groups of type C using affine Hecke algebra of type C. It is still open whether one can generalize this approach to obtain the quantum group analogs for other examples in Section 4, for example, the Ariki–Koike algebras, the Hu algebras or the degenerate affine Hecke algebras.