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

Time:2023-01-13 Fri 17:00-18:30

Venue: Zoom: 815 4690 4797(PW: BIMSA)

Organizer:Nicolai Reshetikhin, Andrey Tsiganov, Ivan Sechin

Speaker:Anton Nazarov Saint Petersburg University


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.

DATEJanuary 13, 2023
Related News
    • 0

      Duality defects from lattice, gauging and symmetry TFT

      AbstractRecent years have witnessed an explosion of studies of the non-invertible symmetries in various dimensions. In this talk, I will revisit the most vanilla type of non-invertible symmetry — Kramers-Wannier duality symmetry in (1+1)d, from three different perspectives: 1. lattice; 2. field theory; 3. Symmetry TFT. I will explain the construction of non-invertible defects, their fusion rul...

    • 1

      Limit sets for branching random walks on relatively hyperbolic groups

      AbstractBranching random walks (BRW) on groups consist of two independent processes on the Cayley graphs: branching and movement. Start with a particle on a favorite location of the graph. According to a given offspring distribution, the particles at the time n split into a random set of particles with mean $r \ge 1$, each of which then moves independently with a fixed step distribution to the ...