Generalised Howe dualityand injectivity of induction: the symplectic case

TL;DR

This paper advances the understanding of symplectic Howe duality by introducing combinatorial methods, generalizing classical results, and proving the injectivity of induction for Levi branchings.

Contribution

It presents two new combinatorial approaches to symplectic Howe duality and proves a conjecture on the injectivity of induction for Levi branchings.

Findings

Established a generalized Howe duality with branching coefficients.

Proved the injectivity of induction for Levi branchings.

Developed combinatorial methods using determinantal formulae and (bi)crystals.

Abstract

We study the symplectic Howe duality using two new and independent combinatorial methods: via determinantal formulae on the one hand, and via (bi)crystals on the other hand. The first approach allows us to establish a generalised version where weight multiplicities are replaced by branching coefficients. In turn, this generalised Howe duality is used to prove the injectivity of induction for Levi branchings as previously conjectured by the last two authors.