Abacus proofs of Schur function identities

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Date
2010
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Journal ISSN
Volume Title
Publisher
Siam Publications
Abstract

This article uses combinatorial objects called labeled abaci to give direct combinatorial proofs of many familiar facts about Schur polynomials. We use abaci to prove the Pieri rules, the Littlewood-Richardson rule, the equivalence of the tableau definition and the determinant definition of Schur polynomials, and the combinatorial interpretation of the inverse Kostka matrix (first given by Egecioglu and Remmel). The basic idea is to regard formulas involving Schur polynomials as encoding bead motions on abaci. The proofs of the results just mentioned all turn out to be manifestations of a single underlying theme: when beads bump, objects cancel.

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Keywords
abaci, schur functions, pieri rules, littlewood-richardson rules, symmetric polynomials, tableaux, inverse kostka matrix, mathematics, applied
Citation
Loehr, N. A., "Abacus proofs of Schur function identities," SIAM J. Discrete Math., 24(4), 1356-1370, (2010). DOI: 10.1137/090753462