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 Lindström-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.

arXiv comment57 pages, 15 figures, 2 tables; v3 fixed typos, added comparison to Biane's result, updated references, fixed typos; v2 fixed typos in Theorem 4.10, 4.14, shorter proof of Theorem 4.6 (thanks to C. Krattenthaler), proved of Conjecture 4.17 in v1

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Cited by in corpus (2)

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• Non symmetric Cauchy kernel, crystals and last passage percolation