On Schützenberger modules of the cactus group

Let $\mathfrak{g}$ be a reductive complex Lie algebra. In Kashiwara’s theory of $\mathfrak{g}$-crystals, the cactus group plays a role analogous to that of the braid group in representations of the quantum group $U_{q}(\mathfrak{g})$. Indeed just as the $n$-strand braid group acts on $n$-fold tensor products of representations of $U_{q}(\mathfrak{g})$ (resulting in a braided category), the cactus group $C_{n}$ acts on $n$-fold tensor products of crystals (resulting in a coboundary category) [HK06]. And just as the type $\mathfrak{g}$ braid group acts on any integrable representation of $U_{q}(\mathfrak{g})$, the type $\mathfrak{g}$ cactus group acts on any $\mathfrak{g}$-crystal [HKRW]. This latter “internal” action is our focus.

Before describing our results, we highlight the appearance of the internal cactus group action in several recent theorems.

Losev construced an action of the cactus group on the Weyl group of $\mathfrak{g}$, and showed that it interacts nicely with Kazhdan-Lusztig cells [LosCacti]. For $\mathfrak{g}=\mathfrak{sl}_{n}$, this recovers the external action of the cactus group corresponding to the zero weight space of the $n$-fold tensor product of the standard representation.

Losev constructs his action by showing that certain wall-crossing functors are perverse equivalences in the sense of Chuang and Rouquier [CRperv]. This was recently extended in work of Halacheva, Losev, Licata and the second author [HLLY]. We show that for any categorical representation of $U_{q}(\mathfrak{g})$, the Rickard complexes corresponding to the half-twist are perverse equivalences. From this we obtain the internal cactus group action on any integrable representation. In [GY, GY2] Gossow and the second author explain how to recover this cactus group action in type A directly from the representation without appealing to categorical techniques.

In a different direction, Halacheva, Kamnitzer, Rybnikov and Weekes study the action of Gaudin algebras on tensor product multiplicity spaces [HKRW]. Their main tool is a crystal structure on eigenvectors for shift of argument subalgebras, which are a family of commutative algebras acting on irreducible $\mathfrak{g}$ representations. In particular they show that the internal action of the cactus group controls the monodromy of these eigenvectors.

In this paper we initiate a study of representations of the cactus group which arise as permutation modules from the internal action on crystals. We’ll now describe our work in detail.

We specialise to the case of $\mathfrak{g}=\mathfrak{sl}_{n}$. The corresponding cactus group $C_{n}$ is an infinite group generated by $c_{J}$, for subintervals $J\subset[1,n]$, subject to the relations in Definition 2.2. It is isomorphic to the orbifold fundamental group of the real locus of the wonderful compactification of $\mathfrak{h}^{reg}/S_{n}$, where $\mathfrak{h}^{reg}$ is the regular locus in the reflection representation of the symmetric group $S_{n}$ [DJS03].

Let $\lambda$ be a partition of $n$. The cactus group $C_{n}$ acts on $\mathrm{SYT}(\lambda)$, the set of standard Young tableau of shape $\lambda$, where the generator $c_{J}$ acts by a partial Schützenberger involution. Letting $S^{\lambda}$ denote the Specht module of $S_{n}$, it is natural to view the $C_{n}$-action on $\mathrm{SYT}(\lambda)$ as an action on the Kazhdan-Lusztig basis of $S^{\lambda}$. We thus obtain a $C_{n}$-action on $S^{\lambda}$, which we term the “Schützenberger module”, and denote $S^{\lambda}_{\mathsf{Sch}}$.

Our main problem, which to our knowledge has not been studied, is the following:

An obvious obstruction to solving this problem is that we do not have a classification of the finite dimensional irreducible representation of $C_{n}$ (it is of wild representation type). Nevertheless, there are naturally occurring families of irreducible representations of $C_{n}$ obtained by inflation from symmetric groups.

Indeed, there is a natural homomorphism $C_{n}\to S_{n}$, and this can be generalised to a surjective map $\pi_{k}:C_{n}\to S_{k}$, for $1\leq k\leq n$ (cf. Lemma 2.3). For $\lambda\vdash k$ we let $S^{\lambda}_{\pi_{k}}$ be the irreducible $C_{n}$-module on $S^{\lambda}$ obtained via pullback by $\pi_{k}$.

Our main theorem solves Problem 1.1 in the case when $\lambda$ is a hook partition. To $\lambda=(a,1^{b})$ a hook partition, we associate a composition of $n-1$ given by $\widetilde{\lambda}=(a-1,b)$.

Note that in the outlying case, $S^{(2,1)}_{\mathsf{Sch}}$ is simply the two-dimensional module with basis elements interchanged by $c_{[1,3]}$ ($c_{[1,2]}$ and $c_{[2,3]}$ act trivially).

Our main tools for proving the theorem come from work of Berenstein and Kirillov [BeKi] and Chmutov, Glick, and Pylyavskyy [CGP]. The former define a group of symmetries of Gelfand-Tsetlin patterns (i.e. semistandard Young tableau), which the latter show is a quotient of the cactus group. These results allow us to show that in the case of a hook shape, the $C_{n}$-action on $S^{\lambda}_{\mathsf{Sch}}$ factors through $S_{n-1}$, and to identify resulting permutation module.

In this section we briefly recall the basic combinatorics of Young tableau. For more details see [Sagan]. Let $n\geq 1$. A partition of $n$, written $\lambda\vdash n$, is a weakly decreasing sequence of nonnegative integers that sum to $n$:

If we drop the weakly decreasing condition, we get the notion of a composition of $n$.

We use Young diagrams to represent partitions and compositions. A Young diagram for a composition $\mu$ is a finite collection of cells, arranged in left-justified rows, where the $i$-th row length is the $i$-th entry of $\mu$.

Let $\lambda\vdash n$. A Young tableau of shape $\lambda$ is a filling of the corresponding Young diagram with positive integers. For example, here is a Young diagram and tableau of shape $\lambda=(5,3,2)$:

The Young tableau is semistandard (respectively standard) if the entries are weakly increasing (respectively strictly increasing) along rows, and strictly increasing down columns. The content of a tableau $T$ of shape $\lambda$ is the composition of $n$, $\mu(T)=(\mu_{1},\mu_{2},\ldots)$, where $\mu_{i}$ is the number of $i$’s appearing in $T$.

Given $\lambda\vdash n$ and $m\geq 1$, we let $\mathrm{SSYT}(\lambda,m)$ denote the set of semistandard Young tableau of shape $\lambda$ and cells filled in with numbers $1,\ldots,m$. We let $\mathrm{SYT}(\lambda)$ denote the set of standard Young tableau of shape $\lambda$ and cells filled in with the numbers $1,\ldots,n$.

The Kostka number $K_{\lambda\mu}$ is defined equivalently as: the number of $T\in\mathrm{SSYT}(\lambda,n)$ of shape $\lambda$ and content $\mu$, the dimension of the $\mu$ weight space in the irreducible representation of $\mathfrak{gl}_{n}$ of highest weight $\lambda$, or as the multiplicity of $S^{\lambda}$ in the permutation module $M^{\mu}$ (Equation (2.1)).

Given partitions $\mu,\lambda$ we write $\mu\subseteq\lambda$ if $\mu_{i}\leq\lambda_{i}$ for every $i$. Let $\lambda,\mu$ be two partitions such that $\mu\subseteq\lambda$. The skew-diagram of shape $\lambda/\mu$ is given by removing the boxes of $\mu$ in $\lambda$. A skew tableau is a labelling of these boxes with positive integers. Here is a skew diagram and tableau of shape $\lambda/\mu$ for $\lambda=(5,5,3)$ and $\mu=(3,2)$:

Similar to above, semistandard tableau on skew shapes are skew-tableau with weakly increasing labels along the rows and strictly increasing labels down the columns. Standard tableau on skew shapes are semistandard tableau whose entries strictly increase along the rows.

Let $\mu$ be a composition of $n$. Let $T,T^{\prime}$ be two diagrams of shape $\mu$ with entries $1,\ldots,n$. We write $T\sim T^{\prime}$ if $T$ and $T^{\prime}$ are row-equivalent, i.e. they have the same entries in each row. An equivalence class for this relation is a $\mu$-tabloid. We let $\mathrm{Tab}(\mu)$ be the set of $\mu$-tabloids. A tabloid can be pictured in a manner similar to tableau. For example, here is a $(3,4)$-tabloid:

representing the equivalence class of the diagram with entries $1,2,3$ in the first row and $4,5,6,7$ in the second.

Set $[a,b]=\{a,a+1,\ldots,b\}$. Given a tableau $T$ we let $T|_{[a,b]}$ be the tableau obtain by deleting all cells with entries not in $[a,b]$.

Let $n\geq 1$. Let $S_{n}$ denote the symmetric group on $\{1,2,\ldots,n\}$. Let $s_{i}\in S_{n}$ denote the simple transposition swapping $i$ and $i+1$. Finite dimensional irreducible complex representations of $S_{n}$ are indexed by partitions $\lambda$ of $n$. The irreducible representation corresponding to $\lambda\vdash n$ is the Specht module $S^{\lambda}$. For instance, $S^{(n)}$ is the trivial representation, $S^{1^{n}}$ is the sign representation, and $S^{(n-1,1)}$ is the standard representation.

The Specht module $S^{\lambda}$ has a remarkable basis indexed by $\mathrm{SYT}(\lambda)$ called the Kazhdan-Lusztig basis, which we denote $\{b_{T}\;\mid\;T\in\mathrm{SYT}(\lambda)\}$. To construct this basis one needs to pass to the Iwahori-Hecke algebra $H_{n}(q)$ associated to $S_{n}$. Kazhdan and Lusztig constructed a canonical basis of the Hecke algebra, which gives rise also to bases of its cell modules. In type A, these cell modules are the irreducible Specht modules and the specialisation $q\mapsto 1$ leads to the basis $\{b_{T}\}$. For more details, see e.g. [KL79, GaMc, Rhoad].

Given a composition $\mu$ of $n$, let $S_{\mu}\subseteq S_{n}$ be the corresponding parabolic subgroup. Let $M^{\mu}$ denote the induced module $\text{Ind}_{S_{\mu}}^{S_{n}}(\mathbb{C})$ from the trivial representation. The module $M^{\mu}$ has a basis indexed by the set of row tabloids $\mathrm{Tab}(\mu)$, where the action is given by permutation of entries. Kostka numbers encode the decomposition of $M^{\mu}$ into Specht modules:

Given an interval $J=[a,b]\subseteq[1,n]$, let $S_{J}\subseteq S_{n}$ be the subgroup of permutations which fix $i\notin J$. In the notation of the previous section, $S_{J}$ is the parabolic subgroup $S_{\mu}$, where $\mu=(1^{a-1},b-a+1,1^{n-b})$. Let $w_{J}\in S_{J}$ be the longest element, that is $w_{J}$ “flips” the interval $[a,b]$ via $a+i\mapsto b-i$ for $0\leq i\leq b-a$.

The cactus group is an infinite group, which has its origins in (a) the study of symmetry groups of universal covers of blow-ups of projective hyperplane arrangements [DJS03], and (b) the study of commutators in the category of crystals for a semisimple Lie algebra [HK06].

Note that there is also a slightly different presentation of the cactus group that’s often used, where generators are indexed by subdiagrams of the Dynkin diagram of a semisimple Lie algebra $\mathfrak{g}$ [HKRW]. The cactus group defined above corresponds to type $A_{n-1}$.

Symmetric groups are naturally quotients of the cactus group. Indeed, we have a map $\pi_{n}:C_{n}\to S_{n},\;c_{J}\mapsto w_{J}$, which is a surjective group homomorphism since the defining relations of $C_{n}$ are satisfied also by the elements $w_{J}\in S_{n}$. This map can be generalised as follows

By inflation we obtain irreducible representations of $C_{n}$ on $S^{\lambda}$, for $\lambda\vdash k$ and $1\leq k\leq n$, which we denote $S^{\lambda}_{\pi_{k}}$.

In order to construct the Schützenberger modules, we need to first recall some operations on Young tableau. For more details see [Sagan].

Let $\lambda\vdash n$. Recall the Kazhdan-Lusztig basis $\{b_{T}\;\mid\;T\in\mathrm{SYT}(\lambda)\}$ of $S^{\lambda}$. We define a $C_{n}$ action on $S^{\lambda}$ using (partial) Schützenberger involutions:

We term the resulting representation the Schützenberger module of $C_{n}$, and denote it by $(\rho_{\lambda},S^{\lambda}_{\mathsf{Sch}})$.

Let $v_{\lambda}=\sum_{T\in\mathrm{SYT}(\lambda)}b_{T}$ and define the $C_{n}$-module $V^{\lambda}$ by the decomposition $S^{\lambda}_{\mathsf{Sch}}=\mathbb{C}v_{\lambda}\oplus V^{\lambda}$. We begin with some preliminary observations about $V^{\lambda}$.

In general, $V^{\lambda}$ is not irreducible. Indeed, let $\delta:\mathrm{SYT}(\lambda)\to\mathrm{SYT}(\lambda^{\prime})$ be the dual map, where the tableau is reflected by the diagonal from northwest to southeast. Here, $\lambda^{\prime}$ is the dual shape of $\lambda$.

The maps $\mathsf{jdt}$ and $\delta$ commute on standard Young tableaux (but not in general). It follows that $\delta$ commutes with the promotion map, and hence the Schützenberger involution. Thus $\delta$ commutes with the $C_{n}$ action on standard Young tableau.

We therefore obtain an isomorphism $\delta:S^{\lambda}_{\mathsf{Sch}}\to S^{\lambda^{\prime}}_{\mathsf{Sch}}$. In particular, for a self-dual shape $\lambda=\lambda^{\prime}$, we have a non-trivial automorphism $\delta:S^{\lambda}_{\mathsf{Sch}}\to S^{\lambda}_{\mathsf{Sch}}$. As $\delta$ is an involution, we have an eigenspace decomposition $S^{\lambda}_{\mathsf{Sch}}=S_{+}^{\lambda}\oplus S_{-}^{\lambda}$ corresponding to the eigenvalues $\pm 1$ of $\delta$:

Notice that $v_{\lambda}\in S_{+}^{\lambda}$, and hence there exists a submodule $W^{\lambda}$ such that $S_{+}^{\lambda}=\mathbb{C}v_{\lambda}\oplus W^{\lambda}$. Thus

and hence $V^{\lambda}=W^{\lambda}\oplus S_{-}^{\lambda}$. Consequently $V^{\lambda}$ is not irreducible for self-dual $\lambda$.

Theorem 1.2 generalises Proposition 3.1 to arbitrary hook-shaped partitions.

In order to undertake a more detailed study of Schützenberger modules we will utilise Gelfand-Tsetlin patterns and their symmetries, as developed by Berenstein and Kirillov.

Let $n\in\mathbb{N}$ and $\lambda\vdash n$. A Gelfand-Tsetlin pattern with $n$ rows and top row $\lambda$ is a triangular arrangement of nonnegative integers $\{\lambda_{i,j}\}_{1\leq i\leq j\leq n}$ such that $\lambda_{i,j+1}\geq\lambda_{i,j}\geq\lambda_{i+1,j+1}$ and the top row is $\lambda$. Denote the set of such patterns as $\mathrm{GTP}(\lambda,n)$.

Define a map $\Phi:\mathrm{SSYT}(\lambda,n)\to\mathrm{GTP}(\lambda,n)$ as follows. Let $T\in\mathrm{SSYT}(\lambda,n)$. For $1\leq k\leq n$, as $T$ is semistandard, $T\big{|}_{[1,k]}$ cannot have more than $k$ rows. Let the shape of $T\big{|}_{[1,k]}$ be $(\lambda_{1,k},\lambda_{2,k},\cdots,\lambda_{k,k})$. As the notation suggests, set $\Phi(T)$ equal to $\mathcal{T}=\{\lambda_{i,j}\}_{1\leq i\leq j\leq n}$. The following is immediate:

Bereinstein and Kirillov defined operators acting on $\mathrm{GTP}(\lambda,n)$ as follows. Let $\mathcal{T}=\{\lambda_{i,j}\}_{1\leq i\leq j\leq n}\in\mathrm{GTP}(\lambda,n)$. Define $\tau_{k}:\mathrm{GTP}(\lambda,n)\to\mathrm{GTP}(\lambda,n)$ for $1\leq k\leq n-1$ by $\tau_{k}(\mathcal{T})=\{\widetilde{\lambda}_{i,j}\}_{1\leq i\leq j\leq n}$, where

For the edge cases we let $a_{1,j}=\lambda_{1,j+1}$ and $b_{j,j}=\lambda_{j+1,j+1}$.

It is conjectured by Berenstein and Kirillov that these generate all relations among the operators $\tau_{1},\tau_{2},\ldots,\tau_{n-1}$.

By transport of structure via $\Phi$, we have an action of $BK_{n}$ on $\mathrm{SSYT}(\lambda,n)$. Following [CGP], we will describe this explicitly. Let $T\in\mathrm{SSYT}(\lambda,n)$. Recall that $T\big{|}_{[k,k+1]}$ is a disjoint union of rectangles and strips of the form

We say a strip is of type $(a,b)$ if it contains $a$ many $\ytableaushort{k}$-boxes and $b$ many $\ytableaushort{\scriptstyle k+1}$-boxes.

Define $\widetilde{\tau}_{k}:\mathrm{SSYT}(\lambda,n)\to\mathrm{SSYT}(\lambda,n)$ such that $\widetilde{\tau}_{k}(T)$ acts on $T\big{|}_{[k,k+1]}$ by replacing each strip of type $(a,b)$ with a strip of type $(b,a)$, and leaving the rectangles unchanged:

In the example above, strips of type $(0,1),(1,0),(1,1),(1,3)$ were swapped with strips of type $(1,0),(0,1),(1,1),(3,1)$.

We define $\widetilde{\tau}_{k}(T)$ to be the tableau obtained by replacing $T\big{|}_{[k,k+1]}$ with $\widetilde{\tau}_{k}(T\big{|}_{[k,k+1]})$, and leaving the other boxes unchanged.