Monday 26 September

I will talk about the usage of the Robinson-Schensted-Knuth (RSK) correspondence to integrable probability focusing on some stochastic processes such as totally asymmetric simple exclusion process (TASEP) and polynuclear growth (PNG) model. One parameter ("q") generalization of this approach will also be briefly addressed. In the context of the symmetric polynomial, it corresponds to the generalization of the Schur polynomial to the q-Whittaker polynomial i.e. the t=0 case of Macdonald (q,t) generalization of the Schur polynomial. By this approach, it becomes possible to analyze the q generalized models for TASEP, PNG etc.

In the last part, I will introduce only a main result of our recent study (arXiv:2106.11922 ) about a new RSK-type approach to the q-models called skew-RSK dynamics, which is a joint work with Matteo Mucciconi and Tomohiro Sasamoto.

Crystal originates in a certain good basis at q=0 of a representation of the quantum group, and from this reason, it gives a strong tool to connect representation theory and combinatorics. In this talk, I concentrate on type A case and (try to) summarize its relation to combinatorics such as Robinson-Schensted correspondence, Kostka polynomial, Kerov-Kirillov-Reshetikhin bijection, and to quantum integrability such as the Yang-Baxter equation, energy function. If time permits, I also mention its super analog.

Quantum groups were introduced in the 80s to describe the mathematical structure responsible for the integrable properties of quantum systems. More recently, the refined notion of "shifted quantum groups" has played an important role in algebraic geometry. This subtle modification of the original definition brings more flexibility in the representation theory. In this talk, I will present several representations for simple shifted quantum groups, namely the shifted quantum affine sl(2) and quantum toroidal gl(1) algebras. Then, I will describe the action of (shifted) quantum groups on families of symmetric polynomials.

I will discuss Feigin-Odesskii shuffle algebras and their connections with integrable models. The main example will be the trigonometric shuffle algebra. This algebra is related to the quantum toroidal algebra of gl_1 and is useful for studying the associated XXZ type integrable model.