Justin Lambright

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

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 $H_n(q)$.