On the action of the Weyl group on canonical bases

TL;DR

This paper investigates how certain elements of simply-laced Weyl groups act on canonical bases in representations, showing they induce bijections up to lower-order terms, with methods from categorical representation theory.

Contribution

It demonstrates that separable elements of Weyl groups act on canonical bases by bijections up to lower-order terms in a broad class of representations, linking combinatorics and categorical methods.

Findings

Separable permutations act on Kazhdan--Lusztig bases via bijections.

Separable elements act on dual canonical bases in tensor products.

Categorical representation theory techniques underpin the analysis.

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.