Total nonnegativity and induced sign characters of the Hecke algebra

TL;DR

This paper develops a combinatorial method to evaluate induced sign characters of the type-A Hecke algebra at specific sums over subgroups and their products, including certain Kazhdan-Lusztig basis elements.

Contribution

It introduces a new combinatorial rule for evaluating induced sign characters of the Hecke algebra at subgroup sums and their products, extending understanding of these algebraic structures.

Findings

Provides explicit evaluation rules for induced sign characters

Includes evaluation at Kazhdan-Lusztig basis elements

Enhances combinatorial tools for Hecke algebra analysis

Abstract

Let $\mathfrak S_{[i,j]}$ be the subgroup of the symmetric group $\mathfrak S_n$ generated by adjacent transpositions $(i,i+1), \dotsc, (j-1,j)$, assuming $1 \leq i < j \leq n$. We give a combinatorial rule for evaluating induced sign characters of the type-$A$ Hecke algebra $H_n(q)$ at all elements of the form $\sum_{w \in \mathfrak S_{[i,j]}} T_w$ and at all products of such elements. This includes evaluation at some elements $C'_w(q)$ of the Kazhdan-Lusztig basis.