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Algebraic and Complex Geometry Seminar
Weyl group symmetries of the toric variety associated with Weyl chambers
Weyl group symmetries of the toric variety associated with Weyl chambers
Organizer
Speaker
Tao Gui
Time
Tuesday, November 19, 2024 3:00 PM - 4:00 PM
Venue
A3-3-301
Online
Zoom 230 432 7880
(BIMSA)
Abstract
For any crystallographic root system, let W be the associated Weyl group, and let WP be the weight polytope (also known as the W-permutohedron) associated with an arbitrary strongly dominant weight. The action of W on WP induces an action on the toric variety X(WP) associated with the normal fan of WP, and hence an action on the rational cohomology ring H^∗ (X(WP)). Let P be the corresponding dominant weight polytope, which is a fundamental region of the W-action on WP. We give a type uniform algebraic proof that the fixed subring H^∗ (X(WP))W is isomorphic to the cohomology ring H^∗(X(P)) of the toric variety X(P) associated with the normal fan of P. I will also explain why our proof applies to non-crystallographic Coxeter groups, in which case there is no toric variety but the “virtual cohomology rings” exist. Joint with Hongsheng Hu and Minhua Liu.
Speaker Intro
I got my Ph. D. in 2023 from the Academy of Mathematics and Systems Science, Chinese Academy of Sciences. Currently I am a postdoc of the Beijing International Center for Mathematical Research, Peking University. My research interests are Lie theory, geometric/combinatorial representation theory, and combinatorial Hodge theory. And I have broad interests in topological, geometric, and combinatorial problems related to representation theory.