Extensions of character formulas by the Littlewood decomposition

TL;DR

This paper generalizes character formulas for symplectic group representations using the Littlewood decomposition, leading to new generating functions and hook length formulas for integer partitions.

Contribution

It introduces new properties of the Littlewood decomposition and applies them to extend existing character formulas and derive novel combinatorial identities.

Findings

Derived signed generating functions for specific integer partition subsets.

Established new hook length formulas related to the Littlewood decomposition.

Extended previous character formulas for symplectic groups.

Abstract

In 2015, the author proved combinatorially character formulas expressing sums of the (formal) dimensions of irreducible representations of symplectic groups, refining some works of Nekrasov and Okounkov, Han, King, and Westbury. In this article, we obtain generalizations of these character formulas, by using a bijection on integer partitions, namely the Littlewood decomposition, for which we prove new properties. As applications, we derive signed generating functions for subsets of integer partitions, and new hook length formulas.