Combinatorial Howe duality of symplectic type

TL;DR

This paper introduces a new combinatorial framework for Howe dual pairs involving symplectic groups and classical Lie (super)algebras, using an analogue of the RSK algorithm and jeu de taquin to unify their representation theory.

Contribution

It develops a symplectic analogue of the RSK algorithm and jeu de taquin for Lie superalgebras, providing a uniform combinatorial interpretation of Howe duality.

Findings

Established a symplectic RSK analogue independent of the Lie algebra type.

Defined a jeu de taquin sliding for spinor models of Lie superalgebras.

Connected the combinatorics to symplectic tableaux of King.

Abstract

We give a new combinatorial interpretation of Howe dual pairs of the form $(\g,{\rm Sp}_{2\ell})$, where $\g$ is a Lie (super)algebra of classical type. This is done by establishing a symplectic analogue of the RSK algorithm associated to this pair, in a uniform way which does not depend on $\g$. We introduce an analogue of jeu de taquin sliding for spinor model of irreducible characters of a Lie superalgebra $\g$ to define $P$-tableau and show that the associated $Q$-tableau is given by a symplectic tableau due to King.