TY - JOUR

T1 - Skew Howe duality and limit shapes of Young diagrams

AU - Nazarov, Anton

AU - Postnova, Olga

AU - Scrimshaw, Travis

PY - 2024/1

Y1 - 2024/1

N2 - We consider the skew Howe duality for the action of certain dual pairs of Lie groups (Formula presented.) on the exterior algebra (Formula presented.) 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 duality for the pairs (Formula presented.), (Formula presented.), (Formula presented.), and (Formula presented.) 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 (Formula presented.) -representation multiplicity is given as a determinant formula using the Lindström–Gessel–Viennot lemma and as a product formula. These admit natural (Formula presented.) -analogs that we show equals the (Formula presented.) -dimension of a (Formula presented.) -representation (up to an overall factor of (Formula presented.)), giving a refined version of the combinatorial skew Howe duality. Using these product formulas (at (Formula presented.)), we take the infinite rank limit and prove that the diagrams converge uniformly to the limit shape.

AB - We consider the skew Howe duality for the action of certain dual pairs of Lie groups (Formula presented.) on the exterior algebra (Formula presented.) 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 duality for the pairs (Formula presented.), (Formula presented.), (Formula presented.), and (Formula presented.) 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 (Formula presented.) -representation multiplicity is given as a determinant formula using the Lindström–Gessel–Viennot lemma and as a product formula. These admit natural (Formula presented.) -analogs that we show equals the (Formula presented.) -dimension of a (Formula presented.) -representation (up to an overall factor of (Formula presented.)), giving a refined version of the combinatorial skew Howe duality. Using these product formulas (at (Formula presented.)), we take the infinite rank limit and prove that the diagrams converge uniformly to the limit shape.

UR - https://www.omu.ac.jp/orp/ocami/assets/21_10.pdf

UR - https://www.sci.osaka-cu.ac.jp/OCAMI/publication/preprint/pdf2021/21_10.pdf

UR - https://www.mendeley.com/catalogue/71a13c20-70fb-3afe-9d31-b1aad81c4b4d/

U2 - 10.1112/jlms.12813

DO - 10.1112/jlms.12813

M3 - Article

VL - 109

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

SN - 0024-6107

IS - 1

M1 - e12813

ER -