On the action of Weyl groups on canonical bases and categorical braid group actions

Abstract: In this talk we'll be considering the following situation: suppose we have a representation $(V,\pi)$ of a Weyl group equipped with a canonical basis. Given an element $g$ of the group, can we extract interesting information about the matrix of $\pi(g)$ with respect to the basis? In general this is extremely difficult but in some situations there are beautiful answers to this question. The first results in this direction are due to Berenstein-Zelevinsky and Stembridge, who proved that the long element of the symmetric group acts on the Kazhdan-Lusztig basis by the Schutzenberger involution on tableau. I will explain vast generalizations of this theorem. The underlying ideas driving these results come from braid group actions on derived categories. This is based on joint work with Martin Gossow.

mathematical physicsalgebraic geometrydifferential geometrygeometric topologyoperator algebrasrepresentation theorysymplectic geometry

Audience: researchers in the topic

Geometry, Physics, and Representation Theory Seminar

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