Bases of the quantum matrix bialgebra and induced sign characters of the Hecke algebra

TL;DR

This paper provides a combinatorial method to evaluate induced sign characters of the type A Hecke algebra at various elements, including Kazhdan-Lusztig basis elements, using transition matrices of the quantum matrix bialgebra.

Contribution

It introduces a novel combinatorial rule for evaluating sign characters of the Hecke algebra, including a subtraction-free formula for basis evaluations.

Findings

First subtraction-free rule for basis evaluation of the H_n(q)-trace space.

Combinatorial description of transition matrices for quantum matrix bialgebra.

Applicable to Kazhdan-Lusztig basis elements indexed by 321-hexagon-avoiding permutations.

Abstract

We combinatorially describe entries of the transition matrices which relate monomial bases of the zero-weight space of the quantum matrix bialgebra. This description leads to a combinatorial rule for evaluating induced sign characters of the type $A$ Hecke algebra $H_n(q)$ at all elements of the form $(1 + T_{s_{i_1}}) \cdots (1 + T_{s_{i_m}})$, including the Kazhdan-Lusztig basis elements indexed by $321$-hexagon-avoiding permutations. This result is the first subtraction-free rule for evaluating all elements of a basis of the $H_n(q)$-trace space at all elements of a basis of $H_n(q)$.