Kazhdan-Lusztig polynomials for 321-hexagon-avoiding permutations

2000/05/05 by Sara Billey, Sara C. Billey, Billey, Sara C. +2 · 1 citation
Mathematics · #05E15 (Primary) 20F55 #14M15 (Secondary) #32S45 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05E15 #msc:14M15 #msc:20F55 #msc:32S45

paper · pdf · doi:10.48550/arxiv.math/0005052

24 pages, 18 figures, AMS-LaTeX

arxiv created 2000/05/05 · openalex publication_date 2000/05/05 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give a combinatorial formula for the Kazhdan-Lusztig polynomials P_x,w in the symmetric group when w is a 321-hexagon-avoiding permutation. Our formula, which depends on a combinatorial framework developed by Deodhar, can be expressed in terms of a simple statistic on all subexpressions of any fixed reduced expression for w. We also show that w being 321-hexagon-avoiding is equivalent to several other conditions, such as the Bott-Samelson resolution of the Schubert variety X_w being small. We conclude with a simple method for completely determining the singular locus of X_w when w is 321-hexagon-avoiding.

Cited by

• A generalization of Deodhar's defect statistic for Iwahori--Hecke algebras of type BC

Related

• Explicit Enumeration of 321,Hexagon-Avoiding Permutations
• 321-polygon-avoiding permutations and Chebyshev polynomials
• Kazhdan--Lusztig polynomials for maximally-clustered hexagon-avoiding permutations
• Embedded factor patterns for Deodhar elements in Kazhdan-Lusztig theory
• Leading coefficients of Kazhdan--Lusztig polynomials for Deodhar elements