Quantized Howe-type dualities via Koornwinder polynomials and the $X=K$ phenomenon

T. Gerber and C. Lecouvey were partially supported by the Agence Nationale de la Recherche funding ANR CORTIPOM 21-CE40-0019. B. Ion was partially supported by the Simons Foundation grant 420882. C. Lenart was partially supported by the NSF grant DMS-2401755.

In this section, we recall some classical results on root systems and the representation theory of the Lie algebras over $\mathbb{C}$. We refer the reader to [1, 8, 15] for a detailed exposition. Consider such a finite-dimensional simple algebra $\mathfrak{g}^{T_{n}}$ with root system $R^{T_{n}}$ of type $T_{n}$, realized in the Euclidean space $E=\oplus_{i=1}^{n}\mathbb{R\varepsilon}_{i}$. When there is no risk of confusion, we will drop the superscript $T_{n}$ to simplify the notation and simply write for example $\mathfrak{g},R$ instead of $\mathfrak{g}^{T_{n}},R^{T_{n}}$. The Dynkin diagram of $R$ is indexed by $I=\{1,\ldots,n\}$ and we denote as usual by

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  $R_{+}$ and $S=\{\alpha_{i},i\in I\}$ the subsets of positive and simple roots respectively,

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  $W$ the Weyl group with generators $s_{i},\;i\in I$ associated with the simple roots $\alpha_{i},i\in I$,

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  $\ell$ the length function on $W$: for any $w$ in $W$, $\ell(w)$ is the number of generators $s_{i}$ in any reduced expression of $w$,

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  $Q$ the root lattice and $Q_{+}$ the cone generated by the positive roots,

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  $P$ the weight lattice and $P_{+}$ the cone of dominant weights, generated by the fundamental weights $\omega_{i},i\in I$,

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  $\rho=\sum_{i=1}^{n}\omega_{i}=\frac{1}{2}\sum_{\alpha\in R_{+}}\alpha$, the half-sum of positive roots,

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  $V(\nu)$ the simple $\mathfrak{g}$-module of highest weight $\nu\in P_{+},$

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  $\leq$ the dominance order on $P$, defined by $\gamma\leq\mu$ if and only if $\mu-\gamma\in Q_{+}$.

We also recall the Weyl character formula. For each dominant weight $\lambda$ in $P_{+}$, the character of $V(\lambda)$ is the polynomial $s_{\lambda}\in\mathbb{Z}^{W}[P]=\{U\in\mathbb{Z}[P]\mid w(U)=U\}$ verifying

We shall also use the notation $a_{\gamma}=\sum_{w\in W}\varepsilon(w)e^{w(\gamma)}$ for any $\gamma\in P$. Then $s_{\lambda}=\frac{a_{\lambda+\rho}}{a_{\rho}}$. The family of polynomials $\{m_{\mu}\mid\mu\in P_{+}\}$ where

is another basis of the character ring $\mathbb{Z}^{W}[P]$. The generalized Kostka numbers are the coefficients in the expansion of the Weyl characters on this basis:

The generalized Kostka number $K_{\lambda,\mu}$ is a nonnegative integer equal to the dimension of the weight space of weight $\mu$ in the representation $V(\lambda)$.

We will be interested in the classical root systems of type $A_{n-1},B_{n},C_{n}$ and $D_{n}$. We will assume the classical realization of these root systems, namely

and

Also, it will be convenient to set $x_{i}=e^{\varepsilon_{i}}$ in order to identify the previous character ring $\mathbb{Z}^{W}[P]$ with the ring of symmetric polynomials for type $A_{n-1}$ in the indeterminates $x_{1},\ldots,x_{n}$ (where we will consider for simplicity the Lie algebra $\mathfrak{gl}_{n}$ rather than $\mathfrak{sl}_{n}$) and with the ring of symmetric Laurent polynomials for types $B_{n}$, $C_{n}$, or $D_{n}$.

Recall here that the affine root systems were classified by Kac (see [19]) in terms of their associated affine Dynkin diagram. Each such Dynkin diagram of type $T_{N}^{(a)}$ is obtained by adding an affine node (usually labelled by $0$) to one of the Dynkin diagrams associated with a finite root system of rank $n$. Here again, in what follows, we will only use a superscript $T_{N}^{(a)}$ when it will be required for the clarity of the exposition. The root lattice so obtained is then

where $\delta^{\prime}=\delta$ is the imaginary null root for any affine classical types but type $A_{2n}^{(2)}$ where $\delta^{\prime}=\frac{1}{2}\delta$. The affine weight lattice can then be described as

where $P=\oplus_{i=1}^{n}\mathbb{Z\omega}_{i}$ is the weight lattice of the finite root system with fundamental weights $\omega_{i},i=1,\ldots n$ and $\Lambda_{0}$ the fundamental affine weight associated with the $0$-node. We will denote by $W_{\mathrm{a}}$ the affine Weyl group associated with our affine root system. It is generated by the affine reflections $s_{i},\>i=0,1,\ldots n$ and contains the finite Weyl group $W$ as the subgroup generated by the $s_{i},i=1,\ldots,n$. In the rest of this paper, we will assume that the affine root systems that we consider are of classical type, that is their underlying finite root system is of type $A,B,C$ or $D$. Kac’s classification ensures that $T_{N}^{(a)}$ is one of $A_{n-1}^{(1)}$, $B_{n}^{(1)}$, $C_{n}^{(1)}$, $D_{n}^{(1)}$ (untwisted types), $A_{2n}^{(2)}$, $A_{2n-1}^{(2)}$, $D_{n+1}^{(2)}$ (twisted types). In fact, we will also need the variations $A_{2n-1}^{(2,{\dagger})}$, $A_{2n}^{(2,{\dagger})}$, and $B_{n}^{(1,{\dagger})}$ of the affine root systems $A_{2n-1}^{(2)}$, $A_{2n}^{(2)}$, and $B_{n}^{(1)}$ respectively, in which we relabel the nodes of the Dynkin diagrams by changing each label $i$ into $n-i$. This does not change the associated root system up to isomorphism but will change the energy statistic. 2 and 4 contain the affine Dynkin diagrams that will be used in this paper.

Recall that crystal graphs can be regarded as combinatorial skeletons of irreducible highest weight modules associated with any simple affine Lie algebra $\widehat{\mathfrak{g}}$, see [13, 2] for generalities. More precisely, in the Kashiwara-Lusztig approach of crystal theory, these objects are defined from representation theory of the quantum group $U_{\nu}(\widehat{\mathfrak{g}}\mathfrak{)}$ associated with $\widehat{\mathfrak{g}}$. The irreducible highest weight $U_{\nu}(\widehat{\mathfrak{g}}\mathfrak{)}$-modules are parametrized by the dominant affine weights: write $V(\Lambda)$ for the module labeled by the dominant weight $\Lambda$. The crystal $B(\Lambda)$ is then an oriented graph with arrows $\overset{i}{\rightarrow}$, where $i\in\{0,1,\ldots,n\}$, equipped with a weight map

One can then compute the character of $V(\Lambda)$ as the generating series of the weight map, that is

This crystal $B(\Lambda)$ has a unique source vertex $b_{\Lambda}$ of weight $\Lambda$ and a natural grading: the degree $\mathrm{d}(b)$ of $b$ in $B(\Lambda)$ is the number of $0$-arrows in any path connecting $b_{\Lambda}$ to $b$. A remarkable property of crystals is its compatibility with the tensor product. More precisely, the decomposition of a tensor product of simple modules into irreducible components is obtained by looking at the decomposition of the associated crystal into its connected components.

There is also a rich finite-dimensional representation theory of a particular subalgebra $U_{\nu}^{\prime}(\widehat{\mathfrak{g}})\subset U_{\nu}(\widehat{\mathfrak{g}})$. The associated simple modules are no longer of highest weight and the associated category is not semisimple. In this article, we are especially interested in some particular simple finite-dimensional such modules called the Kirillov-Reshetikhin modules. These are parametrized by pairs $(r,s)\in\{1,\ldots,n\}\times\mathbb{Z}_{>0}$, and are denoted $B^{(r,s)}$. In what follows, we will only consider these KR modules for $s=1$, which are called column KR modules since $r$ can be interpreted as the height of a column-shaped Young diagram. The KR modules also have associated crystal graphs with $i$-arrows indexed by $\{0,1,\ldots,n\}$. These graphs are finite and connected but do not have a distinguished source vertex (reflecting the fact the KR modules are finite-dimensional but not of highest weight). When $s=1$, they admit a simple combinatorial description in terms of column tableaux introduced by Kashiwara and Nakashima [20], which depends on the classical affine type considered, see [2] for more details.

The tensor products of KR modules are still irreducible, thus the tensor product of column KR crystals give finite-connected crystals. These crystals do not remain irreducible in general when one removes the $0$-arrows. In fact, these $0$-arrows permit to define a subtle statistic $D$, called energy, on any tensor product $B$ of KR crystals in such a way that $D$ is constant on the classical components of $B$ (obtained by removing the $0$-arrows). The definition of $D$ is quite involved and also depends on a choice of normalization (i.e. the choice of a particular classical component where $D$ is zero). For the purpose of this paper, we can in fact bypass this definition by exploiting the connection between our crystal $B$ and a well-chosen affine Demazure crystal, which we will explain in Theorem 2.6. For the moment, let us denote by $\mathrm{HW}(B)$ the set of classical highest weight vertices in $B$, that is the set of vertices in $b$ with no incident $i$-arrows but one $0$-arrow. Each vertex $b$ in $\mathrm{HW}(B)$ admits a classical dominant weight. We can then collect all the classical highest weight vertices in $\mathrm{HW}(B)$ with prescribed dominant weight $\lambda$. The generating series associated with the energy function $D$ over the subset $\mathrm{HW}(B)_{\lambda}$ of $\mathrm{HW}(B)$ with dominant weight $\lambda$ is called a one-dimensional sum (1-d sum for short). In this paper we are particularly interested in the 1-d sums defined from a partition $\mu$ in the rectangular partition $(m^{n})$. (whose Young diagram has $n$ rows and $m$ columns). To such a partition, we can indeed associate the tensor product of column KR crystals

where $\mu_{1}^{\prime},\ldots,\mu_{m}^{\prime}$ are the columns of the Young diagram associated with $\mu$.

In this paragraph, we recall the definition of the Koornwinder polynomials. We refer the reader to [33] for a brief history and more details. Consider a formal parameter $a$ and recall the notation for the $q$-Pochhammer symbol

Similarly for a family of formal parameters $(a_{1},\ldots,a_{m})$ set

In what follows $q,t,a,b,c,d$ are indeterminates and we set $\mathbb{K}=\mathbb{C}(q,t,a,b,c,d)$. Consider the ring $\mathbb{K}^{W}[x_{1}^{\pm 1},\ldots,x_{n}^{\pm 1}]$ of Laurent polynomials in the indeterminates $x_{1},\ldots,x_{n}$ invariant by the action of $W,$ the Weyl group of type $C_{n}$. As mentioned in the previous section, this can be regarded as the character ring of type $C_{n}$ over $\mathbb{K}$. In particular $\mathbb{K}^{W}[x_{1}^{\pm 1},\ldots,x_{n}^{\pm 1}]$ admits various bases indexed by the set $\mathcal{P}_{n}$ of partitions with at most $n$ parts (which here can be identified with the set of dominant weights of type $C_{n}$). We have a natural bar involution on $\mathbb{K}[x_{1}^{\pm 1},\ldots,x_{n}^{\pm 1}]$ such that for any $f$ in $\mathbb{K}[x_{1}^{\pm 1},\ldots,x_{n}^{\pm 1}]$, the polynomial $\overline{f}$ is obtained by replacing each $x_{i}$ by its inverse $x_{i}^{-1}$. We write $[f]_{1}$ for the constant term in $f$. Now we can define a paring on $\mathbb{K}^{W}[x_{1}^{\pm 1},\ldots,x_{n}^{\pm 1}]$ by

where

The following proposition defines the basis of Koornwinder polynomials.

Fix $n,m\in\mathbb{Z}_{\geq 1}$. We denote by $(m^{n})$ the rectangular partition with $n$ rows and $m$ columns. For any partition $\lambda\subseteq(m^{n})$, we set

where $\lambda^{\prime}=(\lambda_{1}^{\prime},\ldots,\lambda_{m}^{\prime})\subseteq(n^{m})$ is the conjugate of $\lambda$.

It is known that these Koornwinder polynomials recover Macdonald polynomials by appropriate specialization of the Askey-Wilson parameters $a,b,c,d$. More precisely, we recover Macdonald polynomials $P_{\lambda}^{T_{N}^{(a)}}(x;q,t,u)$ of each classical affine types $T_{N}^{(a)}$ by using the specializations in Table 4, see e.g. [6].

Observe that the Macdonald polynomials we consider are associated to non-simply laced affine root systems and we can consider the corresponding Macdonald polynomials with unequal parameters $(t,u)$, with $t$ being associated to the roots of square length $2$, and $u$ being associated to roots of square root length $1$ (for $B_{n}^{(1)}$, $B_{n}^{(1,\dagger)}$, $D_{n+1}^{(2)}$), or $4$ (for $A_{2n-1}^{(2)}$, $A_{2n-1}^{(2,\dagger)}$, $C_{n}^{(1)}$), or both (for $A_{2n}^{(2)}$). Note that $A_{2}^{(2)}$ has no roots of square length $2$ and in this case the Macdonald polynomials do not involve the parameter $t$. On the other hand, $D_{n}^{(1)}$ has only roots of square length $2$ and in this case the Macdonald polynomials do not involve the parameter $u$. To streamline the notation we will keep both parameters $t,u$ in the notation even in the situations when the particular Macdonald polynomials depend only one of them.

The more typically used equal parameter Macdonald polynomials $P_{\lambda}^{T_{N}^{(a)}}(x;q,t)$ are obtained by setting $u=t$, that is $P_{\lambda}^{T_{N}^{(a)}}(x;q,t)=P_{\lambda}^{T_{N}^{(a)}}(x;q,t,t)$. The parameter $u$ will allow us to consider the Hall-Littlewood polynomials with unequal parameters which will be considered starting with Section 6.

The Hall-Littlewood polynomials can be regarded as the $q=0$ specializations of Macdonald polynomials. Observe first that by Table 4, once a Macdonald specialization of type $T_{N}^{(a)}$ is performed in a Koornwinder polynomial, the additional $q=0$ specialization only depends on the parabolic subsystem $T_{n}$ of $T_{N}^{(a)}$ obtained by removing its $0$-node. For any dominant weight $\gamma\in P_{+}^{T_{n}}$, we can thus define the Hall-Littlewood polynomial $P_{\gamma}^{T_{n}}(x;t,u)$ by

where $T_{N}^{(a)}$ is any affine root system with underlying classical root system $T_{n}$.

The previous Hall-Littlewood polynomials have a simpler definition independent of Koornwinder-Macdonald polynomial theory. For each classical root system $T_{n}=B_{n}$, $C_{n}$, or $D_{n}$, one can indeed establish that the family of Hall-Littlewood polynomials $\{P_{\gamma}^{T_{n}}(x;u,t)\mid\gamma\in P_{+}^{T_{n}}\}$ yields a basis of the character ring of type $T_{n}$ such that

where the polynomials $K_{\nu,\gamma}(u,t)$ are $(u,t)$-deformations of the generalized Kostka numbers $K_{\nu,\gamma}$ (see (2)). One can show (see [34]) that they can be obtained from the $(u,t)$-Kostant partition function $\mathcal{P}_{u,t}^{T_{n}}$ defined from the series expansion