Bijective proof of a symplectic dual pair identity

Hamel, A.M. and King, R.C. (2011) Bijective proof of a symplectic dual pair identity. SIAM Journal on Discrete Mathematics, 25 (2), 539-560. (doi:10.1137/100802542).

Record type: Article

Abstract

We provide a combinatorial proof of a symplectic character identity relating the sum of a product of symplectic Schur functions to the product $\prod^{m}_{i=1} \prod^{n}_{j=1} (x_i + x^{-1}_{i} + y_j + y_{j}^{-1})$. This formula owes its origin to the existence of a dual pair of symplectic groups acting on spinors, as pointed out by Hasegawa. The first combinatorial proof, based on symplectic tableaux and a variation of the Robinson-Schensted-Knuth correspondence, was due to Terada. Here we use Sch\"utzenberger's jeu de taquin, augmented by two simple zero weight transformations. The identity itself generalizes a well known identity expressing $\prod^{m}_{i=1} \prod^{n}_{j=1} (x_i + y_j)$ as a sum of products of Schur functions that was due to Littlewood and proved combinatorially by Remmel.
We offer an alternative combinatorial proof of this identity by means of the jeu de taquin, as a precursor to the proof of the symplectic identity

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