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Annals of Combinatorics

, Volume 23, Issue 2, pp 367–389 | Cite as

Abacus Proofs of Cauchy Product Identities for Schur Polynomials

  • Nicholas A. LoehrEmail author
Article
  • 26 Downloads

Abstract

The well-known Cauchy Product Identities state that \(\prod _{i,j} (1-x_iy_j)^{-1}= \sum _{\lambda } s_{\lambda }(X) s_{\lambda }(Y)\) and \(\prod _{i,j} (1+x_iy_j)=\sum _{\lambda } s_{\lambda }(X)s_{\lambda '}(Y)\), where \(s_{\lambda }\) denotes a Schur polynomial. The classical bijective proofs of these identities use the Robinson–Schensted–Knuth (RSK) algorithm to map matrices with entries in \({\mathbb {Z}}_{\ge 0}\) or \(\{0,1\}\) to pairs of semistandard tableaux. This paper gives new involution-based proofs of the Cauchy Product Identities using the abacus model for antisymmetrized Schur polynomials. These proofs provide a novel combinatorial perspective on these formulas in which carefully engineered sign cancellations gradually impose more and more structure on collections of matrices.

Keywords

Cauchy product identities Schur polynomials Symmetric functions Antisymmetric functions Involutions 

Mathematics Subject Classification

05E05 05A19 

Notes

Acknowledgements

The author thanks two anonymous referees for their time and their many helpful comments on this paper. In particular, Lemma 2.1 was provided by one of the referees, leading to a simpler proof at the end of Sect. 2.6.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsVirginia TechBlacksburgUSA

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