A combinatorial interpretation of the inverse kostka matrix |
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Authors: | Ömer E[gbar]ecio[gbar]lu Jeffrey B Remmel |
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Institution: | 1. Department of Computer Science , University of California , Santa Barbara, California, 93106;2. Department of Mathematics , University of California-San Diego , La Jolla, California, 92093, California |
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Abstract: | The Kostka matrix K relates the. homogeneous and the Schur bases in the ring of symmetric functions where K λ,μenumerates the number of column strict tableaux of shape λ and type μ. 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=Iis 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. |
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