On the action of the long cycle on the Kazhdan-Lusztig basis

The complex irreducible representations, aka Specht modules, of the symmetric group $S_{n}$ are indexed by partitions of $n$: for such a partition $\lambda\vdash n$ we denote by $S^{\lambda}$ the associated representation. These carry a canonical basis called the Kazhdan-Lusztig basis $\{C_{T}\mid T\in SYT(\lambda)\}$, which is indexed by the set of standard Young tableaux of shape $\lambda$.

Although the Kazhdan-Lusztig basis enjoys many wonderful properties, it is quite complicated to express its transformation under the action of an arbitrary permutation. Nevertheless, owing to a close connection with the RSK correspondence, it is possible for certain distinguished elements.

In the mid 1990s Berenstein-Zelevinsky [1] and Stembridge [10] showed independently that the long element of $S_{n}$ acts on the Kazhdan-Lusztig basis (up to sign) by Schützenberger’s evacuation operator. More recently, Rhoades proved in the case of rectangular partitions that the long cycle $c=(1,2,\ldots,n)$ acts on the Kazhdan-Lusztig basis (up to sign) by the jeu-de-taquin promotion operator, $\mathrm{pr}$ [8]. This was the essential new ingredient needed in Rhoades’ cyclic sieving phenomenon regarding the action of promotion on rectangular standard Young tableaux.

Our first main result is a generalisation of Rhoades’ Theorem to arbitrary partitions. Fix a partition $\lambda\vdash n$ and number its removable boxes $1,2,\dots,r$ moving downward. Define the index of $T\in SYT(\lambda)$ to be the said number of the removable box containing $n$, and denote it $\operatorname{idx}(T)$. Choose an ordering of the Kazhdan-Lusztig basis which is weakly increasing along the index. For $w\in S_{n}$ let $[w]$ be the matrix of $w:S^{\lambda}\to S^{\lambda}$ in this ordered basis.

Equivalently, this theorem states that $c$ acts on the Kazhdan-Lusztig basis by promotion on the leading term, i.e.

where the sum is over $R$ with index strictly smaller than the index of $T$, and $a_{R}\in\mathbf{Z}$. In the case of a rectangular partition, the sum is empty and we recover Rhoades’ Theorem.

The key property of rectangular partitions that Rhoades leverages is that the restriction of the corresponding Specht module to $S_{n-1}$ remains irreducible, and the Kazhdan-Lusztig basis is unchanged if the space is regarded as an $S_{n-1}$-module or an $S_{n}$-module. It is therefore not surprising that the key step in our argument involves the interaction between the restriction functor and the Kazhdan-Lusztig basis in a more general setting, which may be interesting in its own right.

For $k\geq 0$ define the subspace $S^{\lambda}_{k}=\mathrm{span}\{C_{T}\mid T\in SYT(\lambda)\text{ and }\operatorname{idx}(T)\leq k\}$ of $S^{\lambda}$. This gives rise to a filtration $0=S^{\lambda}_{0}\subset S^{\lambda}_{1}\subset S^{\lambda}_{2}\subset\cdots\subset S^{\lambda}_{r}=S^{\lambda}$, where $r$ is the number of removable boxes of $\lambda$.

Note that this theorem provides a new proof that the branching law for symmetric groups is multiplicity-free, as

Finally, we mention a generalisation of Theorem 1.1 to the class of all separable permutations, which includes the long element and the long cycle. By definition, separable permutations are those that can be obtained from $e\in S_{1}$ by repeated applications of direct sum and skew sum. Alternatively, these are the permutations that avoid the patterns $2413$ and $3142$ [6].

Fix $\lambda\vdash n$ an arbitrary partition. In Section 5, we describe briefly how to associate to each separable permutation $w\in S_{n}$ a bijection $\varphi$ of $SYT(\lambda)$ such that an analogous result to Theorem 1.1 holds. That is, if $[w]:S^{\lambda}\to S^{\lambda}$ is the matrix giving the action of $w$ on some given ordering of the KL basis, we have:

In specific cases, we can prove this combinatorially but the general case requires results from categorical representation theory and in particular the action of braid groups on triangulated categories [4]. In our forthcoming work we will describe this connection, and use this to also derive variations of these theorems for arbitrary simply-laced semisimple Lie algebras.

In this section we briefly recall some basic results we’ll need. We refer the reader to [9] for more details on notions from algebraic combinatorics, and to [2] for the necessary background on Kazhdan-Lusztig theory.

In this section we prove Theorem 1.2. Fix $\lambda\vdash n$, and recall the filtration of $S^{\lambda}$ by the subrepresentations $S_{i}^{\lambda}$, each defined as the subspace spanned by $\{C_{T}\}$ with $\operatorname{idx}(T)\leq i$. We will first show that $S_{i}^{\lambda}$ always remains invariant under $S_{n-1}$.

Let $CSS(\lambda)$ be the column super-strict tableau of shape $\lambda$, where the values $1,2,\dots,n$ are placed column-by-column, reading left-to-right. We take $\lambda=(4,3,1)$ as an example:

Preimages of the RSK correspondence when $Q=CSS(\lambda)$ are easy to describe. We simply write the elements of the $P$-tableau in column order, reading from bottom-to-top.

Recall that $\mu_{i}$ is the partition obtained by deleting the $i^{\text{th}}$ removable box from $\lambda\vdash n$. There is a bijection $d_{i}:\{T\in SYT(\lambda)\mid\operatorname{idx}(T)=i\}\to SYT(\mu_{i})$ which removes the largest box of each tableau $T$. By taking its linear extension, we have an isomorphism of vector spaces $d_{i}:S_{i}^{\lambda}/S_{i-1}^{\lambda}\to S^{\mu_{i}}$ for each $1\leq i\leq n-1$. Our goal is to show that this is in fact an isomorphism of $S_{n-1}$-modules. For this, we need to show that the deletion map preserves the function $\overline{\mu}$.

We again define a specific recording tableau and work with the permutations explicitly. For a partition $\lambda\vdash n$ with $r$ removable boxes and $1\leq i\leq r$, define the tableau $CSS_{i}(\lambda)$ by the following procedure:

• •

  Create a tableau of shape $\lambda$, and place the value $n$ in removable box $i$.

• •

  Suppose $n$ is placed in the $k^{\text{th}}$ row. For $1\leq m\leq k-1$, place the value $n-m$ at the end of row $k-m$.

• •

  Place the remaining values $1,\dots,n-k$ into the tableau column by column.

For example, if $\lambda=(4,3,1)$ and $i=2$ we have:

The proof that the $\overline{\mu}$ function is preserved under the deletion map was originally given by Rhoades in the case of rectangular partitions. The following Lemma appears implicitly in the proof:

For a tableau $T\in SYT(\lambda)$ with index $i$, write $[C_{T}]$ for the basis element $C_{T}+S_{i-1}^{\lambda}$ of $S_{i}^{\lambda}/S_{i-1}^{\lambda}$, where $1\leq i\leq n-1$ is the unique value such that $C_{T}\in S_{i}^{\lambda}$, but $C_{T}\notin S_{i-1}^{\lambda}$.

By (2.5), for a tableau $T$ with index $i$ we have:

In the second case, we sum over $R\in SYT(\lambda)$ with $j\in D(R)$ and $\operatorname{idx}(R)=\operatorname{idx}(T)$. Then:

with the sum in the second case as before. If $1\leq j\leq n-2$, then $j\in D(T)$ if and only if $j\in D(d_{i}(T))$, and likewise for $R$. Hence:

This is the action of $s_{j}$ on $C_{d_{i}(T)}$ in $S^{\mu_{i}}$. Thus, $s_{j}$ and $d_{i}$ commute on every basis element $[C_{T}]$ of $S_{i}^{\lambda}/S_{i-1}^{\lambda}$ for every generator $s_{j}$ of $S_{n-1}$. This completes the proof of Theorem 1.2.

In this section we prove Theorem 1.1. Let $w_{n}\in S_{n}$ be the long element, and $w_{n-1}$ be the image of the long element of $S_{n-1}$ under the standard embedding into $S_{n}$. Our goal is to relate the identity $c=w_{n}w_{n-1}$ to $\mathrm{pr}=\mathrm{ev}_{n}\mathrm{ev}_{n-1}$ from Lemma 2.1.

Note that $w_{n-1}\in S_{n-1}$ will act on a tableau of size $n-1$ by $\mathrm{ev}_{n-1}$ (up to sign) [10]. Moreover, $d_{i}$ commutes with $w_{n-1}$ on $S_{i}^{\lambda}/S_{i-1}^{\lambda}$. Therefore

Note that the sign depends only on the shape of $d_{i}(T)$. Since the index of $T$ matches the index of $\mathrm{ev}_{n-1}(T)$, we obtain:

Theorem 1.1 follows immediately from:

In this section we discuss Theorem 1.3. Details will appear in forthcoming work.

Fix a partition $\lambda\vdash n$. Let $I$ denote the set of generators $\{s_{1},\dots,s_{n-1}\}$ of $S_{n}$. For a non-empty subset $J\subseteq I$, let $S_{J}\leq S_{n}$ denote the associated parabolic subgroup, and $w_{J}$ its longest element. To each such subset, we can associate a filtration $0=S_{0}^{\lambda}\subset S_{1}^{\lambda}\subset\cdots\subset S_{p}^{\lambda}=S^{\lambda}$ of the Specht module such that each subspace $S_{j}^{\lambda}$ is $S_{J}$-invariant and spanned by KL basis elements, and the quotient representation $S_{j}^{\lambda}/S_{j-1}^{\lambda}$ is isomorphic to a simple $S_{J}$-module.

This filtration induces a total preorder $\prec_{J}$ on $SYT(\lambda)$, with $T\prec_{J}R$ when $C_{T}$ appears strictly before $C_{R}$ in this filtration. Using partial evacuation operators, we also associate a bijection $\varphi_{J}$ on $SYT(\lambda)$ such that the action of $w_{J}$ on the KL basis is given by

for some constants $a_{R}\in\mathbf{Z}$, and a sign which depends only on the equivalence class of $T$ under $\prec_{J}$.

We can extend this result in a natural way to chains of subsets of $J$. Call a permutation descending if it is of the form $w_{J_{\bullet}}=w_{J_{k}}\cdots w_{J_{1}}$ for some increasing chain $J_{\bullet}$ of subsets $J_{1}\subset\cdots\subset J_{k}\subseteq J$. It turns out that a permutation is descending if and only if it is separable.²²2This observation was communicated to the authors by Joel Gibson, who discovered this fact by enumerating the descending permutations computationally in MAGMA. Each descending permutation has an associated bijection $\varphi_{J_{\bullet}}=\varphi_{J_{k}}\cdots\varphi_{J_{1}}$ and a total preorder $\prec_{J_{\bullet}}$ on $SYT(\lambda)$. The statement is as before, with the action of $w_{J_{\bullet}}$ on the KL basis of $S^{\lambda}$ given by

for some constants $b_{R}\in\mathbf{Z}$, and a sign which depends only on the equivalence class of $T$ under $\prec_{J_{\bullet}}$. Theorem 1.3 follows from this formula.

When $J\subseteq I$ is a connected subset $\{s_{a+1},\dots,s_{b}\}$ for $0\leq a\leq b<n$, then $S_{J}$ is isomorphic to the symmetric group $S_{b-a}$. Moreover, $w_{J}=w_{b+1}w_{b-a}w_{b+1}$, where $w_{k}\in S_{n}$ is the permutation reversing $\{1,\dots,k\}$. Likewise, $\varphi_{J}=\mathrm{ev}_{b+1}\mathrm{ev}_{b-a}\mathrm{ev}_{b+1}$, where $\mathrm{ev}_{k}$ is the partial Schützenberger involution from Section 2.1. Finally, the preorder $\prec_{J_{i}}$ can be defined explicitly by: $R\prec_{J}T$ if and only if $\mathrm{ev}_{a}\mathrm{ev}_{n}(R)$ precedes $\mathrm{ev}_{a}\mathrm{ev}_{n}(T)$ in the total index ordering.

In the case of chains of connected subsets of $J$, equation (5.1) can be proved combinatorially. However, in the case of general chains, we use results from categorical representation theory, which will appear in later work by the authors. This will also prove analogues of these results in other types concerning the action of an ADE Weyl group on the Lusztig’s dual canonical basis in the zero weight space of an irreducible representation of the corresponding Lie algebra.

Finally, we note that the result in (5.1) does not hold for arbitrary permutations. For example, take the non-separable permutation $w=2413\in S_{4}$ and $\lambda=(3,1)$. Choose any of the $3!$ orderings of the KL basis for $S^{\lambda}$, and compute the $QR$ decomposition of $[w]:S^{\lambda}\to S^{\lambda}$ with respect to this basis. One can verify that $Q$ is not a generalised permutation matrix under any of these orderings.