Skew Howe duality, limit shapes of Young diagrams and universal fluctuations

Abstract

Schur-Weyl, Howe and skew Howe dualities in representation theory of groups lead to multiplicity-free decompositions of certain spaces into irreducible representations and can be used to introduce probability measures on Young diagrams that parameterize irreducible representations. It is interesting to study the behavior of such measures in the limit, when groups become infinite or infinite-dimensional. Schur-Weyl duality and GL(n)-GL(k) Howe duality are related to classical works of Anatoly Vershik and Sergey Kerov, as well as Logand-Schepp, Cohn-Larsen-Propp and Baik-Deift-Johannson. Skew GL(n)-GL(k) Howe duality was considered by Gravner, Tracy and Widom, who were interested in the local fluctuations of the diagrams, the limit shapes were studied Sniady and Panova. They demonstrated that results by Romik and Pittel on limit shapes of rectangular Young tableaux are applicable in this case. We consider skew Howe dualities for the actions of classical Lie group pairs: GL(n)-GL(k), Sp(2n)-Sp(2k), SO(2n)-O(2k) on the exterior algebras. We describe explicitly the limit shapes for probability measures defined by the ratios of dimensions and demonstrate that they are essentially the same for all classical Lie groups. Using orthogonal polynomials we prove central limit theorem for global fluctuations around these limit shapes. Using free-fermionic representation we study local fluctuations for more general measures given by ratios of representation characters for skew GL(n)-GL(k) Howe duality. These fluctuations are described by Tracy-Widom distribution in the generic case and in the corner by a certain discrete distribution, first obtained in papers by Gravner, Tracy and Widom. Study of local fluctuations for other classical series remains an open problem, but we present numerical evidence that these distributions are universal. Based on joint works with Dan Betea, Pavel Nikitin, Olga Postnova, Daniil Sarafannikov and Travis Scrimshaw. See arXiv:2010.16383, 2111.12426, 2208.10331, 2211.13728.