Combinatorics of theq-Basis of Symmetric Functions

A basis of symmetric functions, which we denote byqλ(X; q, t), was introduced in the work of Ram and King and Wybourne in order to describe the irreducible characters of the Hecke algebras of type A. In this work we give combinatorial descriptions of the expansions of the functionsqλ(X; q, t) in terms of the classical bases of symmetric functions and apply these results in determining the determinant of the character table of the Iwahori–Hecke algebras and in giving a generating function for certain permutation statistics.