Schur–Weyl duality, Verma modules, and row quotients of Ariki–Koike algebras

We consider the tensor product of a parabolic universal Verma module with the $m$-folded tensor product of the standard representation for $\mathcal{U}_{q}(\mathfrak{gl}_{m})$ to establish a Schur–Weyl duality with a quotient of Ariki–Koike algebras. Ariki–Koike algebras were first considered by Cherednik in [10] as a cyclotomic quotient of the affine Hecke algebra of type $A$. These algebras were later rediscovered and studied by Ariki and Koike [3] from a representation theoretic point of view. Independently, Broué and Malle attached in [7] a Hecke algebra to certain complex reflection groups, and Ariki–Koike algebras turn out to be the Hecke algebras associated to the complex reflection groups $G(d,1,n)$.

Recall that the Ariki–Koike algebra $\mathcal{H}(d,n)$ with parameters $q\in\Bbbk^{*}$ and $\underline{u}=(u_{1},\ldots,u_{d})\in\Bbbk^{d}$ is the $\Bbbk$-algebra with generators $T_{0},T_{1},\ldots T_{n-1}$, where $T_{1},\ldots T_{n-1}$ generate a finite-dimensional Hecke algebra of type $A$ and $T_{0}$ satisfies $T_{0}T_{1}T_{0}T_{1}=T_{1}T_{0}T_{1}T_{0}$, $T_{0}T_{i}=T_{i}T_{0}$ for $i>1$, and $\prod_{i=1}^{d}(T_{0}-u_{i})=0$. We consider the semisimple case, where the simple modules $V_{\mu}$ of $\mathcal{H}(d,n)$ are indexed by $d$-partitions of $n$.

Let $\underline{m}=(m_{1},\ldots,m_{d})$ be a $d$-tuple of positive integers and $\mathcal{P}^{n}_{\underline{m}}$ be the set of all $d$-partitions $\mu=(\mu^{(1)},\ldots,\mu^{(d)})$ of $n$ such that $l(\mu^{(i)})\leq m_{i}$ for all $1\leq i\leq d$.

In this paper we introduce the row-quotient algebra $\mathcal{H}_{\underline{m}}(d,n)$, that depends on $\underline{m}$ as the quotient of $\mathcal{H}(d,n)$ by the kernel of the surjection

Let $M^{\mathfrak{p}}(\Lambda)$ be a parabolic Verma module and $V$ the standard representation for $\mathcal{U}_{q}(\mathfrak{gl}_{m})$. In our conventions, $\mathfrak{p}$ is standard and has Levi factor $\mathfrak{l}=\mathfrak{gl}_{m_{1}}\times\cdots\times\mathfrak{gl}_{m_{d}}$, with $m_{i}\geq 1$ and $m_{1}+m_{2}+\dotsm+m_{d}=m$ and $\Lambda$ depends on $d$ algebraically independent parameters $\lambda_{1},\ldots,\lambda_{d}$ (see Section 3.2 for more details). Thanks to the braided structure on the category of integrable modules over $\mathcal{U}_{q}(\mathfrak{gl}_{m})$, we define a left action of $\mathcal{H}(d,n)$ on $M^{\mathfrak{p}}(\Lambda)\otimes V^{\otimes n}$ in Section 4. Our main result is:

The isomorphism in Equation (1) has several particular specializations (Corollaries 4.3-4.7), some of them recovering well-known algebras:

• •

  If $\mathfrak{p}=\mathfrak{gl}_{m}$ and $m\geq n$, then $\operatorname{End}_{\mathcal{U}_{q}(\mathfrak{gl}_{m})}(M^{\mathfrak{p}}(\Lambda)\otimes V^{\otimes n})$ is isomorphic to the Hecke algebra of type $A$.

• •

  If $\mathfrak{p}=\mathfrak{gl}_{m}$ and $m=2$, then $\operatorname{End}_{\mathcal{U}_{q}(\mathfrak{gl}_{m})}(M^{\mathfrak{p}}(\Lambda)\otimes V^{\otimes n})$ is isomorphic to the Temperley–Lieb algebra of type $A$.

• •

  For $\mathfrak{p}$ such that $m\geq nd$ and $m_{i}\geq n$ for all $1\leq i\leq d$, then $\operatorname{End}_{\mathcal{U}_{q}(\mathfrak{gl}_{m})}(M^{\mathfrak{p}}(\Lambda)\otimes V^{\otimes n})$ is isomorphic to the Ariki–Koike algebra $\mathcal{H}(d,n)$.

• •

  If $\mathfrak{p}$ is such that $d=2$ and $m_{1},m_{2}\geq n$, then $\operatorname{End}_{\mathcal{U}_{q}(\mathfrak{gl}_{m})}(M^{\mathfrak{p}}(\Lambda)\otimes V^{\otimes n})$ is isomorphic to the Hecke algebra of type $B$ with unequal and algebraically independent parameters (see [14, Example 5.2.2, (c)]).

• •

  If the parabolic subalgebra $\mathfrak{p}$ coincides with the standard Borel subalgebra of $\mathcal{U}_{q}(\mathfrak{gl}_{m})$ then $\operatorname{End}_{\mathcal{U}_{q}(\mathfrak{gl}_{m})}(M^{\mathfrak{p}}(\Lambda)\otimes V^{\otimes n})$ is isomorphic to Martin–Woodcock’s [20] generalized blob algebra $\mathcal{B}(d,n)$. This generalizes the case of $\mathcal{U}_{q}(\mathfrak{sl}_{2})$ covered in [15].

In the last case, this gives a new interpretation of the generalized blob algebras $\mathcal{B}(d,n)$ in terms of Schur–Weyl duality. We also give a new presentation of $\mathcal{B}(d,n)$ as a quotient of Ariki–Koike algebras:

The idea of writing this note originated when we started thinking of possible extensions of our work in [18] to more general Kac–Moody algebras and were not able to find the appropriate generalizations of [15] in the literature. When we were finishing writing this note Peng Shan informed us about [23], whose results are far beyond the ambitions of this article. Nevertheless, we expect our results to be connected to [23, §8] using a braided equivalence of categories between a category of modules for the quantum group $\mathcal{U}_{q}(\mathfrak{gl}_{m})$ and a category of modules over the affine Lie algebra $\widehat{\mathfrak{gl}}_{m}$, which is due to Kazhdan and Lusztig [17]. However, the explicit description of the endomorphism algebra of $M^{\mathfrak{p}}(\Lambda)\otimes V^{\otimes n}$, which was our first motivation towards categorification later on, does not seem to appear anywhere in [23] except in the particular case of our 4.5.

Another motivation for the results presented here resides in the potential applications to low-dimensional topology, as indicated in [22]. We find that it would be also interesting to investigate the use of several Verma modules in a tensor product as suggested in [11].

We would like to thank Steen Ryom-Hansen for comments on an earlier version of this paper. The authors would also like to thank the referee for his/her numerous, detailed, and helpful comments. The authors were supported by the Fonds de la Recherche Scientifique - FNRS under Grant no. MIS-F.4536.19.

Fix once and for all a field $\Bbbk$ and two positive integers $d$ and $n$ and choose elements $q\in\Bbbk^{*}$ and $u_{1},\ldots,u_{d}\in\Bbbk$. We recall the definition of the Ariki–Koike algebra introduced in [3], which we view as a quotient of the group algebra of the Artin–Tits braid group of type $B$.

As in the type $A$ Hecke algebra, for any $w\in\mathfrak{S}_{n}$ we can define unambiguously $T_{w}$ by choosing any reduced expression of $w$.

It is shown in [3] that the algebra $\mathcal{H}(d,n)$ is of dimension $d^{n}n!$ and a basis is given in terms of Jucys–Murphy elements, which are recursively defined by $X_{1}=T_{0}$ and $X_{i+1}=T_{i}X_{i}T_{i}$.

We end this section with a semisimplicity criterion due to Ariki [2], which in our conventions takes the following form.

In this section, we suppose that the algebra $\mathcal{H}(d,n)$ is semisimple. In [3], Ariki and Koike gave a construction of the simple $\mathcal{H}(d,n)$-modules, using the combinatorics of multipartitions.

We now define the row quotients of $\mathcal{H}(d,n)$ which will appear later as endomorphism algebras of a tensor product of modules for $\mathcal{U}_{q}(\mathfrak{gl}_{m})$.

Similar to the case of $\mathcal{H}(d,n)$, we have inclusions $\mathcal{H}_{\underline{m}}(1,d)\subset\mathcal{H}_{\underline{m}}(2,d)\subset\cdots\subset\mathcal{H}_{\underline{m}}(n,d)$ and the branching rule is governed by the corresponding truncation of the Young lattice of $d$-partitions.

Let $q$ be an indeterminate. The following definition of $\mathcal{U}_{q}(\mathfrak{gl}_{m})$ is over the field $\mathbb{Q}(q)$, but, via scalar extension, we will also consider it over a field containing $\mathbb{Q}(q)$ without further notice.

We endow it with a structure of a Hopf algebra, with comultiplication $\Delta$, counit $\varepsilon$ and antipode $S$ given on generators by the following:

Set $\mathcal{U}_{q}(\mathfrak{gl}_{m})^{0}$ as the subalgebra generated by $(L_{i})_{1\leq i\leq m}$, and $\mathcal{U}_{q}(\mathfrak{gl}_{m})^{\geq 0}$ as the subalgebra generated by $(L_{i},E_{j})_{\begin{subarray}{c}1\leq i\leq m\\ 1\leq j\leq m-1\end{subarray}}$.

We denote by $P=\bigoplus_{i=1}^{m}\mathbb{Z}\varepsilon_{i}$ the weight lattice of $\mathfrak{gl}_{m}$ with $\mathbb{Z}$-basis given by the fundamental weights $(\varpi_{i})_{1\leq i\leq m}$ where $\varpi_{i}=\varepsilon_{1}+\cdots+\varepsilon_{i}$. We denote by $Q$ the root lattice with $\mathbb{Z}$-basis given by the simple roots $(\alpha_{i})_{1\leq i\leq d-1}$ where $\alpha_{i}=\varepsilon_{i}-\varepsilon_{i+1}$. Denote by $\Phi^{+}$ the set of positive roots, by $P^{+}$ the set of dominant weights for $\mathfrak{gl}_{m}$, that is $\mu=\sum_{i=1}^{m}\mu_{i}\varepsilon_{i}$ with $\mu_{1}\geq\mu_{2}\geq\cdots\geq\mu_{m}$. We also endow $P$ with the standard non-degenerate bilinear form: $\langle\varepsilon_{i},\varepsilon_{j}\rangle=\delta_{i,j}$. The symmetric group $\mathfrak{S}_{m}$ acts on $P$ by permuting the coordinates and leaves the bilinear form $\langle\cdot,\cdot\rangle$ invariant. Finally, let $\rho$ be the half-sum of the positive roots.

We will often work with extensions $\mathbb{Z}[\beta_{1},\ldots,\beta_{k}]\otimes_{\mathbb{Z}}P$, where the $\beta_{i}$’s are indeterminates and we also extend the bilinear form $\langle\cdot,\cdot\rangle$ to $\mathbb{Z}[\beta_{1},\ldots,\beta_{k}]\otimes_{\mathbb{Z}}P$.

Suppose that our field $\Bbbk$ contains the field $\mathbb{Q}(q)$ and let $M$ be an $\mathcal{U}_{q}(\mathfrak{gl}_{m})$-module over the ground field $\Bbbk$. An element $v\in M$ is said to be a weight vector if $L_{i}v=\varphi(\varepsilon_{i})v$, where $\varphi\colon P\rightarrow\Bbbk$ is the corresponding weight. The module $M$ is said to be a weight module if the action of the elements $L_{1},\ldots,L_{m}$ is simultaneously diagonalizable. A highest weight module is a weight module $M$ such that $M=\mathcal{U}_{q}(\mathfrak{gl}_{m})v$, where $v$ is a weight vector such that $E_{i}v=0$ for $1\leq i\leq m-1$.

It is well-known that finite dimensional weight $\mathcal{U}_{q}(\mathfrak{gl}_{m})$-modules of type $1$ are parameterized by the set $P^{+}$ of dominant weights.

In this paper, we will be interested in modules over the field $\mathbb{Q}(q,\lambda_{1},\ldots,\lambda_{k})$, where $\lambda_{i}=q^{\beta_{i}}$ is an indeterminate (recall that $q$ is formal and so $q^{\beta_{i}}$ is also formal). Moreover, we only consider type $1$ modules, where the weights are of the form

for some $\mu\in\mathbb{Z}[\beta_{1},\ldots,\beta_{k}]\otimes_{\mathbb{Z}}P$ and for all $\nu\in P$.

We now turn to parabolic Verma modules. Let $\mathfrak{p}$ be a standard parabolic subalgebra of $\mathfrak{gl}_{m}$ with Levi factor $\mathfrak{l}=\mathfrak{gl}_{m_{1}}\times\cdots\times\mathfrak{gl}_{m_{d}}$, where $m_{i}\geq 1$ and $\sum_{i=1}^{d}m_{i}=m$. Denote by $I$ the set $\left\{\tilde{m}_{i}\ |\ 1\leq i\leq d-1\right\}$, where $\tilde{m}_{i}=m_{1}+\ldots+m_{i}$, so that $\mathcal{U}_{q}(\mathfrak{l})$ is generated by $L_{i},E_{j}$ and $F_{j}$ for $1\leq i\leq m$ and $j\not\in I$ and $\mathcal{U}_{q}(\mathfrak{p})$ is generated by $L_{i},E_{j}$ and $F_{k}$ for $1\leq i\leq m$, $1\leq j\leq m-1$ and $k\not\in I$. Denote by $P^{+}_{i}$ the set of dominant weights for $\mathfrak{gl}_{m_{i}}$. We identify the set $P^{+}_{1}\times\cdots\times P^{+}_{d}$ with the dominant weights $P^{+}_{\mathfrak{l}}$ of $\mathfrak{l}$ by the following map