Tuesday 27 September

The skew RSK dynamics is a deterministic dynamics on the space of pairs of semi-standard tableaux (P,Q) recently introduced by Imamura, Sasamoto and myself. Like the box-ball system it displays solitonic behavior and its symmetries can be described in terms of a novel type A bi-crystal structure on the set of tableaux (P,Q). Leveraging symmetries we are able to linearize the dynamics, implementing an inverse scattering method. This appears to be different than other linearizations known in literature, such as the KKR correspondence.

Based on a collaboration with T. Imamura and T. Sasamoto.

Macdonald polynomials are a remarkable family of symmetric functions that are known to have connections to combinatorics, algebraic geometry and representation theory. Due to work of Corteel, Mandelshtam and Williams, it is known that they are related to the asymmetric simple exclusion process (ASEP) on a ring.

The modified Macdonald polynomials are obtained from the Macdonald polynomials using an operation called plethysm. It is natural to ask whether the modified Macdonald polynomials are related to some other particle system. In this talk, we answer this question in the affirmative via a multispecies totally asymmetric zero-range process (TAZRP). We also present a Markov process on tableaux that projects to the TAZRP and derive formulas for stationary probabilities and certain correlations. We also prove a remarkable symmetry property for local correlations.

This is joint work with Olya Mandelshtam and James Martin.

We show that the celebrated six-vertex model of statistical mechanics (along with its multistate generalizations) can be reformulated as an Ising-type model with only a two-spin interaction. Such a reformulation unravels remarkable factorization properties for row to row transfer matrices, allowing one to uniformly derive all functional relations for their eigenvalues and present the coordinate Bethe ansatz for the eigenvectors for all higher spin generalizations of the six-vertex model. The possibility of the Ising-type formulation of these models raises questions about the precedence of the traditional quantum group description of the vertex models. Indeed, the role of a primary integrability condition is now played by the star-triangle relation, which is not entirely natural in the standard quantum group setting, but implies the vertex-type Yang-Baxter equation and commutativity of transfer matrices as simple corollaries. As a mathematical identity the emerging star-triangle relation is equivalent to the Pfaff-Saalschuetz-Jackson summation formula, originally discovered by J. F. Pfaff in 1797. Plausibly, all vertex models associated with quantized affine Lie algebras and superalgebras can be reformulated as Ising-type models. (Based on the joint work with Sergey Sergeev, arXiv:2205.10708)

The box-ball system is an integrable cellular automaton on a one-dimensional lattice. It arises from either quantum or classical integrable systems by the procedures called crystallization and ultradiscretization, respectively. I will explain our point of view of the box-ball systems as an application of the theory of crystals, mainly based on the review paper coauthored with R. Inoue and A. Kuniba [J. Phys. A: Math. Theor. 45 (2012) 073001].