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Topics in Representation Theory
Affine Bruhat order, Kazhdan–Lusztig combinatorial invariance, and spooky dualities
Affine Bruhat order, Kazhdan–Lusztig combinatorial invariance, and spooky dualities
演讲者
Gaston Burrull
时间
2025年12月17日 13:00 至 14:30
地点
A14-201
线上
Zoom 242 742 6089
(BIMSA)
摘要
I will present experimental discoveries on the Bruhat order of affine Weyl groups, revealing surprisingly rigid combinatorial structure. In joint work with Libedinsky and Villegas, we show that all Bruhat intervals in A2 tilde are determined by a simple convex-geometric construction. More strongly, we give a complete classification of dominant intervals: two intervals are poset-isomorphic if and only if their associated polygons are congruent. This produces a striking equivalence between Euclidean geometry and the combinatorics of affine Bruhat intervals.
Our proof implies invariance of Kazhdan–Lusztig polynomials for all intervals in this classification, and it suggests that the long-standing Lusztig–Dyer combinatorial invariance conjecture might hold for unexpectedly simple reasons. For the intervals we classify—and, we conjecture, for every affine Weyl group—every nontrivial isomorphism between Bruhat intervals arises as a composition of a global symmetry, an isomorphism coming from the finite Weyl group, and a piecewise translation. Moreover, the full structure of an interval is already encoded in its dihedral subintervals, hinting at a simple mechanism underlying invariance.
Finally, I will describe a "spooky" phenomenon: certain intervals turn out to be isomorphic to the dual of others, sharing the same R-polynomial, suggesting the action of a non-existent "phantom" longest element in affine Weyl groups.
Our proof implies invariance of Kazhdan–Lusztig polynomials for all intervals in this classification, and it suggests that the long-standing Lusztig–Dyer combinatorial invariance conjecture might hold for unexpectedly simple reasons. For the intervals we classify—and, we conjecture, for every affine Weyl group—every nontrivial isomorphism between Bruhat intervals arises as a composition of a global symmetry, an isomorphism coming from the finite Weyl group, and a piecewise translation. Moreover, the full structure of an interval is already encoded in its dihedral subintervals, hinting at a simple mechanism underlying invariance.
Finally, I will describe a "spooky" phenomenon: certain intervals turn out to be isomorphic to the dual of others, sharing the same R-polynomial, suggesting the action of a non-existent "phantom" longest element in affine Weyl groups.