Tamsen Whitehead

The combinatorics of transition matrices between the bases of the symmetric functions and the B n analogues

Abstract In this paper, we develop the combinatorial interpretations of the transition matrices between the bases of the Bn analogues of the symmetric functions. In order to provide a complete reference, we have included a summary of the transition matrices for symmetric functions as well.

Combinatorics of theq-Basis of Symmetric Functions

A basis of symmetric functions, which we denote byq?(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 functionsq?(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.

Transition matrices and kronecker product expansions of symmetric functions

The Kronecker product of two homogeneous symmetric polynomials P1 and P2 is defined by means by the Frobenius map by the formula . By using the combinatorial definitions for the transition matrices between the various bases of the space of homogeneous symmetric functions, one obtains a procedure for expanding this Kronecker product. In this paper, we employ this technique to study this expansion for the monomial symmetric functions.