Workshop Kirillov–75. Combinatorics and
Bethe ansatz.
November 26–27, 2023

This talk is focused on quantum integrable systems on a classical background. In physics such systems are known as Born-Oppenheimer approximations, when heavy atoms are classical and electrons are quantum. In mathematics, perhaps, most known structures of this type are Azumaya algebras (an algebra that is finite dimensional over the center) and quantum groups at roots of unity. After the description of general mathematical framework several natural examples will be given, such as spin chains, spin Calogero-Moser systems and isomonodromic deformations. The talk is based on joint work with A. Liashyk and I. Sechin.

I will speak about a new combinatorial description for stable Grothendieck polynomials and Lascoux polynomials in terms of cellular decompositions of Gelfand-Zetlin polytopes. This generalizes an earlier result on key polynomials (aka characters of Demazure modules) by Kiritchenko, Timorin and myself. The talk is based on a joint work with Ekaterina Presnova.  

For Szego polynomials on the unit circle we present explicit examples of bispectrality which makes these polynomials similar to "classical" orthogonal polynomials. These examples admit extension to much wider class of special Baxter polynomials. Affine and double affine Hecke algebras of rank 1 arise naturally in this approach from first principles.  

In this talk we introduce a universal weight system (a function on chord diagrams satisfying the 4-term relation) taking values in the ring of polynomials in infinitely many variables, whose particular specialisations are weight systems associated with the Lie algebras gl(N) and Lie superalgebras gl(M|N). We extend this weight system to permutations and provide an efficient recursion for its computation.

I explain what are the fermionic formulas and why they are interesting and important and present some relatively new results — fermionic formulas related with  triplet-like vertex algebras.

In the late 1990s motivated by a question of V.Arnold the speaker and M.Shapiro have studied the algebra generated by the curvature forms of the standard linear bundles over the space of complete flags in C^n. This was the first example of the so-called external zonotopal algebra associated to the complete graph K_n. Since then a number of modifications and generalizations of this algebra defined for all undirected graphs has been introduced. I will briefly survey the field many advances in which were inspired by suggestions and ideas of Anatol Kirillov. 

In this talk, I will introduce the research of lattice walk in analytic combinatorics. Starting from simple one dimensional discrete random walks, I will show how algebraic structures affect the the solution. The result in two dimensional walks is most attracting. We will meet many concepts such as algebraic curves, conformal mapping and Riemann surface in solving two dimensional walks. In the last part of this talk, I will talk about the relation between lattice walk and integrable phase model.