On the inverse Kostka matrix

A combinatorial interpretation of the inverse t-Kostka matrix

In this paper we use tournament matrices to give a combinatorial interpretation for the entries of the inverse t-Kostka matrix, which is the transition matrix between the Hall-Littlewood polyno-mials and the Schur functions. 0.1 Introduction In the rst section of this paper we introduce some basic notation about tournament matrices, and prove a theorem that is crucial in the second section. In ...

From quasisymmetric expansions to Schur expansions via a modified inverse Kostka matrix

Every symmetric function f can be written uniquely as a linear combination of Schur functions, say f = ∑ λ xλsλ, and also as a linear combination of fundamental quasisymmetric functions, say f = ∑ α yαQα. For many choices of f arising in the theory of Macdonald polynomials and related areas, one knows the quasisymmetric coefficients yα and wishes to compute the Schur coefficients xλ. This paper...

Polynomiality of the Q, T-kostka Revisited

Let K(q, t) = K λµ (q, t) λ,µ be the Macdonald q, t-Kostka matrix and K(t) = K(0, t) be the matrix of the Kostka-Foulkes polynomials K λµ (t). In this paper we present a new proof of the polynomiality of the q, t-Kostka coefficients that is both short and elementary. More precisely, we derive that K(q, t) has entries in Z[q, t] directly from the fact that the matrix K(t) −1 has entries in Z[t]....

On the nonnegative inverse eigenvalue problem of traditional matrices

In this paper, at first for a given set of real or complex numbers $sigma$ with nonnegative summation, we introduce some special conditions that with them there is no nonnegative tridiagonal matrix in which $sigma$ is its spectrum. In continue we present some conditions for existence such nonnegative tridiagonal matrices.

An Elementary Method for Computing the Kostka Coefficients