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 $H_n(q)$.