Abstracts

Maud De Visscher (City London) 

Graded representations of Temperley-Lieb algebras

In this talk I will discuss the representations of ordinary Temperley-Lieb algebras, one-boundary TL algebras (also known as blob algebras) and two-boundary TL algebras (also known as symplectic blob algebras). Plaza and Ryom-Hansen obtained a grading on the ordinary and the one-boundary TL algebras by relating these to quiver Hecke algebras (introduced by Khovanov-Lauda and Rouquier). The corresponding graded decomposition numbers are given by Kazhdan-Lusztig polynomials. I will review these results and then explain how we can obtain a grading for the two-boundary case using the orientifold quiver Hecke algebra introduced by Varagnolo-Vasserot and formulate a conjecture for the graded decomposition numbers. This is joint work with Chris Bowman, Zajj Daugherty, Rob Muth and Loic Poulain D’Andecy. [video]

Misha Feigin (Glasgow) 

Quasi-invariant Hermite polynomials and automorphisms of Cherednik algebras 

After reviewing some basic facts about Hermite polynomials I would like to discuss their generalisations stemming from Calogero-Moser theory at integer coupling parameter. This can be done both in one dimension and in many dimensions leading to multivariate polynomials in the latter case. These polynomials are eigenfunctions of the Calogero-Moser quantum Hamiltonian in the harmonic confinement in a suitable guage and they satisfy certain quasi-invariance conditions. This approach also allows to define some "higher versions" of these quasi-invariant multivariable Hermite polynomials by making use of automorphisms of rational Cherednik algebras. The talk is based on a joint work with M. Hallnas and A.P. Veselov. [pdf] [video] 

Anton Izosimov (Glasgow)

Geometry of Goncharov-Kenyon systems

Goncharov-Kenyon integrable systems are completely integrable systems arising from the dimer model on a torus. In this talk, we will review their construction and discuss their geometric interpretation. This is based on joint work with P. Pylyavskyy. https://arxiv.org/pdf/2512.15499

Duncan Laurie (Edinburgh) 

Representations of quantum toroidal algebras 

Quantum toroidal algebras are the ‘double affine’ objects within the quantum world. Their principal module category Ô is the natural toroidal analogue of the finite-dimensional modules for quantum affine algebras. After first discussing the structure of these quantum toroidal algebras, we shall outline some more recent results on their representation theory. These include a well-defined tensor product and monoidal structure on Ô, and a meromorphic braiding by R-matrices which solve the Yang-Baxter equation. Time permitting, I’ll also mention work in progress with Théo Pinet exhibiting special subcategories of Ô as monoidal categorifications of cluster algebras in type A. [pdf] [video]

Alexander Shapiro (Edinburgh)

Coulomb branches and cluster varieties

Given a quiver Г, Braverman, Finkelberg, and Nakajima define a non-commutive algebra A_Г known in physics as the quantised K-theoretic Coulomb branch of the quiver gauge theory determined by Г. Spectra of semi-classical limits of these algebras provide phase spaces of many known integrable systems, including open Toda, Ruijsenaars, and Gelfand–Tsetlin ones. In this talk, for Г without 1-cycles I will construct quivers Q(Г), such that the quantised algebra of global functions on the corresponding cluster variety is isomorphic to A_Г. I will also discuss some applications of this cluster structure to integrable systems. This is a joint work with Gus Schrader, written in the most recent version of the preprint https://arxiv.org/pdf/1910.03186

Alexander Cooper (Heriot-Watt) 

Pre-seminar on Quantum Groups 

Drinfeld-Jimbo quantum groups are important algebraic structures in the theory of quantum integrable systems.  They arise naturally via deformations of Kac-Moody Lie algebras.  In particular, they admit the structure of a (quasitriangular) Hopf algebra. In this pre-seminar, we will recall the definition of a Hopf algebra, and review how the existence of a special element in their twofold tensor products, the universal R-matrix, leads to the construction of solutions to the Yang-Baxter equation. [pdf] [video]

 Yegor Zenkevich (Edinburgh) 

Quantum toroidal algebras and physics

Quantum toroidal algebras are quantum deformations of double loop algebras. They have some remarkable properties and rich representation theory. I will briefly introduce quantum toroidal algebras and explain their relations to integrable systems, gauge theories and string theory. [video]

Francesco Dilda (Glasgow) 

q-deformed Haldane-Shastry model and its crystal limit 

The Haldane-Shastry model is an integrable spin-chain with long-range interaction which can be seen as a long-range version of the Heisenberg XXX model. As for the Heisenberg model, we can deform it by introducing an anisotropy parameter q. For q generic, it retains many of the remarkable properties of the isotropic case such as higher Hamiltonians, symmetry under an action of quantum group). This talk will mainly focus on what happens in the limit q → ∞, the crystal limit. In particular, I will present the current open problems of the crystal limit of this model. This is joint work with Jules Lamers. [pdf] [video]

Theresa Ortscheidt (Glasgow) 

Hecke Algebras at Roots of Unity 

The Iwahori-Hecke algebra of type A can be seen as a q-deformation of the symmetric group algebra. At generic values of q its representation theory is similar to that of the symmetric group and can be understood combinatorially in terms of Young diagrams and Littlewood-Richardson coefficients. However, when q is a root of unity, the Hecke algebra is no longer semi-simple. Goodman and Wenzl introduced a ring in terms of a particular class of Hecke representations at roots of unity, which can be realised as a quotient of the ring of symmetric polynomials.  I will discuss briefly some of the open problems associated with finding a new combinatorial description of the structure coefficients of this ring. [pdf] [video]