Skew Howe duality and limit shapes of Young diagrams

TL;DR

This paper explores skew Howe duality for certain Lie group pairs, providing combinatorial interpretations via lattice paths and tilings, deriving formulas for multiplicities, and analyzing the asymptotic limit shapes of Young diagrams.

Contribution

It establishes a combinatorial framework for skew Howe duality using crystal bases and lattice paths, and analyzes the asymptotic limit shapes of Young diagrams in this context.

Findings

Derived determinant and product formulas for multiplicities.

Established q-analog formulas matching q-dimensions of representations.

Proved convergence of Young diagram shapes to a limit shape in the infinite rank limit.

Abstract

We consider the skew Howe duality for the action of certain dual pairs of Lie groups $(G_1, G_2)$ on the exterior algebra $\bigwedge(\mathbb{C}^{n} \otimes \mathbb{C}^{k})$ as a probability measure on Young diagrams by the decomposition into the sum of irreducible representations. We prove a combinatorial version of this skew Howe for the pairs $(\mathrm{GL}_{n}, \mathrm{GL}_{k})$, $(\mathrm{SO}_{2n+1}, \mathrm{Pin}_{2k})$, $(\mathrm{Sp}_{2n}, \mathrm{Sp}_{2k})$, and $(\mathrm{Or}_{2n}, \mathrm{SO}_{k})$ using crystal bases, which allows us to interpret the skew Howe duality as a natural consequence of lattice paths on lozenge tilings of certain partial hexagonal domains. The $G_1$-representation multiplicity is given as a determinant formula using the Lindstr\"om-Gessel-Viennot lemma and as a product formula using Dodgson condensation. These admit natural $q$-analogs that we show equals the $q$-dimension of a $G_2$-representation (up to an overall factor of $q$), giving a refined version of the combinatorial skew Howe duality. Using these product formulas (at $q =1$), we take the infinite rank limit and prove the diagrams converge uniformly to the limit shape.