Skew Howe duality and limit shapes of Young diagrams

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.