We will discuss solvable lattice models in tandem with the representation theory of Heisenberg algebras and the associated \tau-functions. The boson-fermion correspondence is a Heisenberg algebra isomorphism between two representations, one a Clifford algebra module called fermionic Fock space and the other living on the algebra of symmetric functions and called the bosonic Fock space. Under this isomorphism, certain group-like operators map basis vectors to functions, called \tau-functions, which are eigenfunctions of differential hierarchies called soliton equations. This striking relationship was first studied by the Kyoto school in the early 1980s and has remained a fruitful area of research ever since.
We will focus on the connections between this story and that of solvable lattice models. Both are integrable systems in the sense that they are (in some sense) exactly solvable, but have substantial combinatorial and representation theoretic differences. Despite these differences, \tau-functions and their generalizations in some cases coincide with the partition functions of solvable lattice models, and these encompass several families of particularly interesting functions, such as Schur functions, Whittaker functions, and LLT polynomials.
Ben Brubaker (Minnesota)
Solvable lattice models satisfy Yang-Baxter equations (YBEs). The familiar train argument, making repeated use of YBEs, that swaps adjacent rows of the lattice often gives rise to a Hecke algebra action on the partition function. We survey various examples in recent literature of this phenomenon, where the Hecke action manifests as certain divided difference operators acting on the partition function of the lattice model. In particular, we'll explain how colored lattice models (as appearing in the work of Borodin, Wheeler and collaborators) have partition functions that realize families of non-symmetric refinements of familiar symmetric functions. This talk will contain joint work with various collaborators, including Valentin Buciumas, Dan Bump, and Henrik Gustafsson.
Valentin Buciumas (Amsterdam)
The representation theory of quantum groups is a rich source of solvable lattice models that satisfy the Yang-Baxter equation. In this mostly introductory talk, I will explain how one connects the two notions and survey some recent examples of solvable lattice models and how they are related to certain quantum groups.
In the second part of the talk, I will explain U-turn solvable lattice models and how they connect to quantum symmetric pairs.
Farrokh Atai (Leeds)
The deformed Macdonald-Ruijsenaars (MR) models were introduced as mathematically natural generalizations of the MR models that share many of the mathematical properties of the MR models. In particular, the deformation of the trigonometric Ruijsenaars-Macdonald model (corresponding to A-type root systems) also has polynomial eigenfunctions given by the super-Macdonald polynomials. In this talk, I will review some results on the deformed MR model and the super-Macdonald polynomials, and show that the super-Macdonald polynomials are orthogonal with respect to a natural generalization of Macdonald's inner product.
Based on joint work with M. Hallnäs and E. Langmann: Comm. Math. Phys. 388, 435–468 (2021)
Paul Zinn-Justin (Melbourne)
After reviewing recent developments in the field of Schubert calculus, we'll describe two new ``puzzle rules'' -- combinatorial rules for computing products of Schubert classes
(i.e., [double] Schubert or Grothendieck polynomials) and more generally of Segre motivic classes.
This is joint work with Allen Knutson.
As an elliptic generalization of the non-stationary Ruijsenaars functions proposed by Shiraishi, we introduce a non-Kerov deformation of the Macdonald polynomials.
Geometrically it is motivated by the generating function of the elliptic genera of the affine Laumon spaces and can be related to the partition function of a six dimensional gauge theory. The talk is based on joint works with H.Awata, A.Mironov and A.Morozov.
Junichi Shiraishi (Tokyo)
The aim of my talk is to recall present understandings and several conjectures concerning the so-called non-stationary Ruijsenaars functions, obtained by the collaboration with Edwin Langmann and Masatoshi Noumi. I want to organize materials in a pedagogical manner as much as possible. One should note that many ideas actually have come and have been grown from the quantum Toda or q-Toda equations. Therefore considerable part of the talk is devoted to giving accounts of the degeneration scheme for the non-stationary Ruijsenaars functions in which we have the Toda or q-Toda equa- tions, and the other stuffs including the Schur functions, the Macdonald functions, the irreducible affine characters, and the stationary Ruijsenaars functions.