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

2025/06/25 by Hobbs, Gavin, Parisi, Tommy, Skandera, Mark +1
#05E10 #20C08 #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2506.20099

Abstract

Let H be the Iwahori--Hecke algebra corresponding to any Coxeter group. Deodhar's defect statistic [Geom. Dedicata 36, (1990) pp.95--119] allows one to expand products of simple Kazhdan--Lusztig basis elements of H in the natural basis of H. Clearwater and the third author gave a type-A extension [Ann. Comb. 25, no. 3 (2021) pp.757--787] of this formula which combinatorially describes the natural expansion of products of Kazhdan--Lusztig basis elements indexed by smooth elements of the symmetric group. We similarly give a type-BC extension of Deodhar's result which combinatorially describes the natural expansion of Kazhdan--Lusztig basis elements indexed by hyperoctahedral group elements which are simultaneously smooth in types B and C.

Citations

• Hyperoctahedral group characters and a type-BC analog of graph coloring
• Bases of the quantum matrix bialgebra and induced sign characters of the\n Hecke algebra
• Evaluations of Hecke algebra traces at Kazhdan-Lusztig basis elements
• Kazhdan-Lusztig polynomials for 321-hexagon-avoiding permutations

Related

• On Iwahori--Hecke algebras with unequal parameters and Lusztig's isomorphism theorem
• The character values of Iwahori--Hecke algebras on Coxeter basis elements
• Defect in cyclotomic Hecke algebras
• Relative Kazhdan--Lusztig cells
• Leading coefficients and cellular bases of Hecke algebras