Séminaire Lotharingien de Combinatoire, 93B.90 (2025), 12 pp.
Minh-Tâm Quang Trinh and Nathan Williams
Cell Decompositions of Hecke Traces and Link Polynomials
Abstract.
For specific trace functions on the Hecke algebra of a finite Weyl group W, we establish formulas for their values at positive braids in terms of point counts of Deodhar-type cells in associated algebraic varieties.
For irreducible W, we deduce a uniform enumeration result that interpolates between its rational Catalan and parking combinatorics, generalizing our earlier work with Galashin and Lam.
The key is a new relationship between the varieties from that work and the braid Steinberg varieties introduced by Trinh.
For W = Sn, we prove a similar point-counting formula for each a-degree in the HOMFLYPT polynomial of the link closure of the braid, generalizing work of Shende-Treumann-Zaslow for the "highest" degree.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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