[zoom channel 933-433-492, password=the order of Symmetric group S_6]

23 Aug

10:00-10:40 A. M. Vershik. К 75-летию С.В.Керова (1946-2000) и 50-летию Асимптотической Теории Представлений (AТП -- начало 70-х гг.)

11:00-11:40 F. Petrov. Maxwell--Poincare lemma for Stiefel manifold, and Wishart measures

12:00-12:40 Yu. A. Neretin. Groups of infinite matrices over finite fields

14:00-14:40 I. Pak. Hook formulas and their generalizations

15:00-15:40 N. Reshetikhin. On tensor powers of representations of quantum groups at roots of unity

24 Aug

10:00-10:40 L. Petrov. Free fermion six vertex model and symmetric rational functions

11:00-11:40 N. Tsilevich. The Schur–Weyl graph and Thoma’s theorem

14:00-14:40 G. Koshevoy. Schur positivity and cluster mutations

15:00-15:40 V. Kaimanovich. Coincidence, equivalence and singularity of harmonic measures

16:00-16:40 A. Knizel. Stationary measure for the open KPZ equation

17:00 - memorial session

25 Aug

10:00-10:40 A. Nazarov. Skew Howe duality and limit shape of Young diagrams

11:00-11:40 V. Evtushevsky. Ergodicity of Kerov--Goodman measures on Young--Fibonacci graph

12:00-12:40 A. Malyutin. Growth in groups and the number of knots

**Abstracts**

I. Pak

Starting with the classical hook-length formula (HLF), I will give a broad overview of the many hook formulas across the literature. I will then discuss our latest beta-deformation of the HLF, which arises from the study of factorial Grothendieck polynomials. The talk will be completely combinatorial and assumes no prior knowledge of the area. Joint work with Alejandro Morales and Greta Panova.

L. Petrov

I'll talk about a new generalization of Schur polynomials and related determinantal measures on domino tilings. These constructions arise from the free fermion six vertex model. The domino weights in the model depend on 4 families of inhomogeneity parameters. We discuss a bulk limit which leads to a new inhomogeneous kernel generalizing the discrete sine kernel. Based on a joint work with Amol Aggarwal, Alexei Borodin, and Michael Wheeler.

Я расскажу об обобщении многочленов Шура и детерминантных мер на замощениях доминошками, которые получаются из свободно-фермионной шестивершинной модели. Получающаяся модель замощений доминошками содержит 4 семейства неоднородных параметров. Мы описываем предел "в гуще", который приводит к новым детерминантным ядрам типа двумерного дискретного синус-ядра. Совместная работа с А.Аггарвалом, А.Бородиным, и М.Уилером.

A. Knizel

The Kardar-Parisi-Zhang (KPZ) equation is the stochastic partial differential equation that models stochastic interface growth. In the talk I will present the construction of a stationary measure for the KPZ equation on a bounded interval with general inhomogeneous Neumann boundary conditions. Along the way, we will encounter classical orthogonal polynomials, the asymmetric simple exclusion process, and precise asymptotics of q-Gamma functions. This is a joint work with Ivan Corwin.

N. Tsilevich

We introduce a graded graph, called the Schur–Weyl graph, which is a

covering of the Young graph closely related to the RSK algorithm. Then

we explain how it allows one to obtain a new, purely combinatorial,

proof of Thoma's theorem, which describes the indecomposable

characters of the infinite symmetric group, in the discrete case.

Based on joint work with A.M.Vershik.

V. Kaimanovich

In the absence of measures fully invariant with respect to a group action,

this role can be to a certain extent played by the measures "invariant on

average", with respect to a certain fixed distribution on the group. These

measures are called stationary, and they naturally arise as harmonic

measures of random walks. I will provide several partial answers to the

general question about the dependence of stationary measures on the

underlying step distributions on the group. The talk is based on joint

work with Behrang Forghani.

A. Nazarov

We consider skew Howe duality that is related to the action of a pair

of Lie groups on the exterior algebra

$\bigwedge(\mathbb{C}^{n}\otimes\mathbb{C}^{k})$. This exterior

algebra admits a multiplicity-free decomposition into a direct sum of

tensor products of representations for the pairs of dual groups

$(GL_{n},GL_{k})$, $(SO_{n},Pin_{k})$ and $(Sp_{n},Sp_{k})$. From the

point of view of a single group in a pair this is a tensor power

decomposition into the sum of the irreducible representations. Such a

decomposition can be used to introduce a probability measure on Young

diagrams parameterizing the representations that appear in the sum. In

the limit of infinite rank of the groups the diagrams converge to a

limit shape. In order to derive the limit shape we need to express the

dimension of the dual group representation in terms of the original

diagram and thus obtain the explicit formula for the multiplicities of

tensor power decomposition into the sum of the irreducibles. We

connect this multiplicity to the number of paths in the lozenge tiling

of a certain hexagonal domain and use the Lindström–Gessel–Viennot

lemma, as well as a recursion of the determinants to prove the product

formulas for the multiplicities. These paths are naturally connected

to the crystals for the corresponding representations. We then compute

the asymptotic of the probability measure and derive the explicit

formula for the limit shapes of Young diagrams, that is different from

the results of A.M. Vershik, S.V. Kerov and P. Biane. We discover that the

limit shapes of Young diagrams for the symplectic and orthogonal

groups are ``halves'' of the limit shape of the general linear group.