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On the action of the Weyl group on canonical bases

Fern Gossow, Oded Yacobi

TL;DR

The paper establishes that separable elements of simply-laced Weyl groups act on canonical bases such as the Kazhdan–Lusztig basis and dual canonical bases in tensor products by bijections up to lower-order terms, with the action described via crystal-theoretic involutions and preorders. It does so by embedding the problem in categorical representation theory, using Rickard complexes to produce perverse equivalences that categorify Weyl-group actions and yield combinatorial maps on bases. The work clarifies and unifies known phenomena for the symmetric group (evacuation, promotion) and extends them to broader Weyl groups and tensor-product contexts, including a detailed analysis for Specht modules and Levi branching. It further provides new tools, such as nested evacuation and generalized Rhoades-type results, to understand how separable elements interact with canonical bases in a wide range of representations. The results have potential implications for crystal combinatorics, categorification, and the study of symmetry actions in representation theory.

Abstract

We study representations of simply-laced Weyl groups which are equipped with canonical bases. Our main result is that for a large class of representations, the separable elements of the Weyl group $W$ act on these canonical bases by bijections up to lower-order terms. Examples of this phenomenon include the action of separable permutations on the Kazhdan--Lusztig basis of irreducible representations for the symmetric group, and the action of separable elements of $W$ on dual canonical bases of weight zero in tensor product representations of a Lie algebra. Our methods arise from categorical representation theory, and in particular the study of the perversity of Rickard complexes acting on triangulated categories.

On the action of the Weyl group on canonical bases

TL;DR

The paper establishes that separable elements of simply-laced Weyl groups act on canonical bases such as the Kazhdan–Lusztig basis and dual canonical bases in tensor products by bijections up to lower-order terms, with the action described via crystal-theoretic involutions and preorders. It does so by embedding the problem in categorical representation theory, using Rickard complexes to produce perverse equivalences that categorify Weyl-group actions and yield combinatorial maps on bases. The work clarifies and unifies known phenomena for the symmetric group (evacuation, promotion) and extends them to broader Weyl groups and tensor-product contexts, including a detailed analysis for Specht modules and Levi branching. It further provides new tools, such as nested evacuation and generalized Rhoades-type results, to understand how separable elements interact with canonical bases in a wide range of representations. The results have potential implications for crystal combinatorics, categorification, and the study of symmetry actions in representation theory.

Abstract

We study representations of simply-laced Weyl groups which are equipped with canonical bases. Our main result is that for a large class of representations, the separable elements of the Weyl group act on these canonical bases by bijections up to lower-order terms. Examples of this phenomenon include the action of separable permutations on the Kazhdan--Lusztig basis of irreducible representations for the symmetric group, and the action of separable elements of on dual canonical bases of weight zero in tensor product representations of a Lie algebra. Our methods arise from categorical representation theory, and in particular the study of the perversity of Rickard complexes acting on triangulated categories.
Paper Structure (31 sections, 37 theorems, 87 equations, 1 figure)

This paper contains 31 sections, 37 theorems, 87 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Set-up
  3. The symmetric group
  4. Other Weyl groups
  5. Methodology
  6. Organisation of paper
  7. Representations, crystals and categorifications
  8. Basic objects
  9. Crystals
  10. Categorical representation theory
  11. Perverse equivalences
  12. Definition and properties
  13. Prior results
  14. Perversity for separable elements
  15. Combinatorial applications
  16. ...and 16 more sections

Key Result

Theorem 1.4

Fix $w \in S_n^{\mathrm{sep}}$ and $\lambda \vdash n$ arbitrary. Then $w$ acts on $(S^\lambda,\mathbb{K}\mathbb{L}_\lambda)$ by a bijection up to lower-order terms.

Figures (1)

  • Figure 1: The crystal graph for $\mathbb{L}_4(2,2)$ with arrow labels $\color{lightgray}{\xrightarrow{1}}$, $\color{blue}{\xrightarrow{2}}$ and $\color{red}{\xrightarrow{3}}$. If $J=I_{[2,4]}=\{2,3\}$, then $\mathsf{Res}_J\mathbb{L}_4(2,2)$ has three connected components. The standard tableaux are separately coloured, and are fixed by $\xi_I$.

Theorems & Definitions (86)

  • Definition 1.2
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 2.1
  • Definition 3.3
  • Lemma 3.4
  • proof
  • Theorem 3.5: Corollary 6.7 in HLLY
  • ...and 76 more