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arXiv · 1705.06437

Shifted tableaux and products of Schur's symmetric functions

Keiichi Shigechi
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Abstract

We introduce a new combinatorial object, semistandard increasing decomposition tableau and study its relation to a semistandard decomposition tableau introduced by Kraśkiewicz and developed by Lam and Serrano. We also introduce generalized Littlewood--Richardson coefficients for products of Schur's symmetric functions and give combinatorial descriptions in terms of tableau words. The insertion algorithms play central roles for proofs. A new description of shifted Littlewood--Richardson coefficients is given in terms of semistandard increasing decomposition tableaux. We show that a "big" Schur function is expressed as a sum of products of two Schur $P$-functions, and vice versa. As an application, we derive two Giambelli formulae for big Schur functions: one is a determinant and the other is a Pfaffian.

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Keiichi Shigechi. 2017-05-18. Shifted tableaux and products of Schur's symmetric functions. https://arxiv.org/abs/1705.06437

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