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

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 $$H_n(q)$$ .