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We combinatorially describe entries of the transition matrices which relatemonomial bases of the zero-weight space of the quantum matrix bialgebra. Thisdescription leads to a combinatorial rule for evaluating induced signcharacters 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-Lusztigbasis elements indexed by $321$-hexagon-avoiding permutations. This result isthe 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 $H_n(q)$.

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