Abacus Proofs of Cauchy Product Identities for Schur Polynomials

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.