Linear and Multilinear Algebra, 26 (1990), pp. 59-84.
Ömer Egecioglu and Jeffrey B. Remmel
Combinatorial Interpretation of the Inverse Kostka Matrix
Abstract.
The Kostka matrix $K$ relates the homogeneous and the Schur bases
in the ring of symmetric functions where $K_{\lambda ,\mu}$ enumerates
the number of column strict tableaux of shape $\lambda$ and type $\mu$.
We make use of the Jacobi-Trudi identity to give a combinatorial
interpretation for the inverse of the Kostka matrix in terms of certain
types of signed rim hook tabloids. Using this interpretation, the matrix
identity $ KK^{-1} = I$ is given a purely combinatorial proof. The
generalized Jacobi-Trudi identity itself is also shown to admit a
combinatorial proof via these rim hook tabloids. A further application of
our combinatorial interpretation is a simple rule for the evaluation of a
specialization of skew Schur functions that arises in the computation of plethysms.
omer@cs.ucsb.edu