Affine Bruhat order, Kazhdan-Lusztig invariance, and unexpected dualities

Geometric Representation Theory Seminar

Organizers：

Lin Chen, Will Donovan, Penghui Li, Peng Shan, Changjian Su, Wenbin Yan

Speaker：

Gaston Burrull (BICMR)

Time：

Fri., 10:30-11:30 am, Oct. 31, 2025

Venue：

B627, Shuangqing Complex Building

Title:

Affine Bruhat order, Kazhdan-Lusztig invariance, and unexpected dualities

Abstract:

I will present experimental discoveries on the Bruhat order of affine Weyl groups, revealing intriguing combinatorial structure. In joint work with Libedinsky and Villegas, we classified thick dominant Bruhat intervals in type A2 tilde, showing that each poset is determined by the isometry class of a certain polygon, providing a strong bridge between Bruhat order and Euclidean geometry.

These results suggest that the Lusztig-Dyer combinatorial invariance conjecture for Kazhdan-Lusztig polynomials may hold for surprisingly simple reasons. We conjecture that all nontrivial isomorphisms of affine Bruhat intervals appear only as global poset isomorphisms or piecewise local translations, and all the information of a Bruhat interval is captured by its dihedral subintervals.

I also observed the remarkable phenomenon that some intervals are isomorphic to the dual of others, as if a non-existent phantom longest element exists in affine Weyl groups.