# Darstellungstheorietage and Nikolaus Conference 2021

Speaker: Thomas Gerber (Lausanne)

Title: Combinatorial Howe Duality

Abstract:

The Howe duality is a classical result in Lie theory, which can be expressed as an identity between weight multiplicities for $$\mathfrak{g}_m$$-representations on the one hand and fundamental tensor multiplicities for $$\mathfrak{g}_n$$-representations on the other hand. Here, $$m$$ and $$n$$ are fixed integers and $$\mathfrak{g}$$ is a simple complex Lie algebra of fixed classical type ($$A$$, $$B$$, $$C$$ or $$D$$).

In this talk, I will present a bijective proof of the above identity in the type $$A$$ and $$C$$ cases, relying on the combinatorics of tableaux and crystals. More precisely, weight multiplicities are counted by certain tableaux and fundamental tensor multiplicities are counted by sources in certain crystal graphs which we can explicitely put into one-to-one correspondence.

This is joint work with Jérémie Guilhot and Cédric Lecouvey.