Second Workshop on Lie Theory and Integrability
December 14-15, 2024

Lascoux polynomials simultaneously generalize several important families of polynomials arising in algebraic combinatorics, geometry and representation theory: Schur polynomials (aka characters of irreducible GL(n)-modules, or representatives of Schubert cycles in the cohomology ring of a Grassmannian), key polynomials (characters of Demazure modules in GL(n)-modules) and, finally, symmetric Grothendieck polynomials, obtained as representatives of structure sheaves of Schubert varieties in the K-group of a Grassmannian.
I will speak about several combinatorial interpretations of Lascoux polynomials, due to Tianyi Yu (using set-valued Young tableaux and Kohnert tableaux) and Ekaterina Presnova and myself. The latter interpretation describes Lascoux polynomials as weighted sums over subsets of a certain cellular decomposition of Gelfand–Zetlin polytope. This generalizes our earlier result on key polynomials (joint with Valentina Kiritchenko and Vladlen Timorin). Time permitting, I will say a few words on (mostly conjectural) generalizations of these results to the case of symplectic groups.

In this talk, we explain how the crystal structures of quantum groups and various combinatorics of Young tableaux, like Schensted bumping operator, Littlewood– Richardson rule, Schutzenberger involution, can be obtained from the singular perturbation of linear ordinary differential equations at a second order pole.

A Brauer pair of a pair of non-isomorphic finite groups together with bijections between their conjugacy classes and irreducible characters which respect characters values and power map. A tuple of finite groups is called a Bruaer tuple if two of them form a Brauer pair. In this talk we present a construction of Bruaer tuple using the Jacquet-Langlands correspondence of p-adic linear reductive groups. This is a joint work with Professor Jiukang Yu (Chinese University of Hong Kong).

We derive quantum generalised spin Calogero–Moser systems from the representations of rational and trigonometric Cherednik algebras. Our construction is based on the reducible polynomial representations of Cherednik algebras. Namely, we consider parabolic ideals in the polynomial representation of Cherednik algebra, which turn out to be invariant under the whole Cherednik algebra action for specific values of parameters of Cherednik algebra. Computing the second order Weyl-invariant polynomial in Dunkl operators in factor representation we obtain an explicit formula for the generalised spin Calogero– Moser operator, which acts in the representation of the Weyl group of Cherednik algebra. For the particular case of reflection representation we obtain the operators acting as reflections in deformed root system previously considered by Chalykh, Goncharenko and Veselov. Based on joint work in progress with Misha Feigin, Christian Korff and Martin Vrabec.

Capping operator is one the core subject in the K-theoretic quasimap counting to quiver varieties. It has been shown by Okounkov and Smirnov that it satisfies a system of qdifference equations governed by the MO quantum affine algebras. In this talk we will show how to construct the similar quantum difference equation via the shuffle algebras. Then we will show how to use the monodromy data of these quantum difference equations to prove the isomorphism of the positive half of the MO quantum affine algebras of affine type A and the positive half of the quantum toroidal algebras. If time permits, I will also give a brief explanation on how to extend the proof to the general case.

Considering the ring of ordinary differential operators D_1=K[[x]][d] as a subring of a certain complete non-commutative ring $\hat{D}_1$ (not the known ring of formal pseudodifferential operators!), the normal forms of differential operators mentioned in the title are obtained after conjugation by some invertible operator («Schur operator»), calculated using one of the operators in a ring. Normal forms of commuting operators are polynomials with constant coefficients in the differentiation, integration and shift operators, which have a finite order in each variable, and can be effectively calculated for any given commuting operators. We’ll talk about some applications of the theory of normal forms: generic properties of normal forms and of Schur operator, an effective parametrisation of torsion free sheaves with vanishing cohomologies on aprojective curve, characterisation of differential operators.

Double Affine Hecke Algebras were originally introduced by I.Cherednik and used in his 1995 proof of Macdonald conjecture from algebraic combinatorics. These algebras come equipped with a large automorphism group SL(2,Z) which has geometric origin, namely it is the modular group of a torus. It was subsequently shown that spherical Double Affine Hecke Algebras realize universal flat deformations of the quantum chracter variety of a torus and their existence is closely related to the fact that classical SL(n,C)-character varieties admit symplectic resolution of singularities via the Hilbert Scheme Hilb_n(\mathbb

C*\times\mathbb C*). In 2019 G.Belamy and T.Schedler have shown that SL(n,C)-character varieties of closed genus g surface admit symplectic resolutions only when g=1 or (g,n)=(2,2). In my talk I will discuss our (g,n)=(2,2) generalization of Double Affine Hecke Algebra which provide a flat deformation of quantum SL(2,C)-character variety of a closed genus two surface. I will show that solution to the word problem in our algebra has striking similarity with the Poicare–Birkhoff–Witt Theorem for the basis of Universal Enveloping Algebra of a Lie algebra. This is consistent with the philosophy formulated by A.Okounkov that resolutions of symplectic singularities should be viewed as "Lie Algebras of the XXI'st century".

(joint with Sh. Shakirov)

The spacetime monopole equation is an interesting hyperbolic integrable system which is a dimension reduction of the anti-self-dual Yang-Mills equation on R^{2,2}. I will explain some results for this integrable system, including inverse scattering method and Backlund transformations. This talk is based on a joint work with Chuu-Lian Terng and Karen Uhlenbeck.

Toda lattice is an integrable lattice model describing motions of a chain of particles with exponential interactions between nearest neighbors. Since 1967 after its discovery, Toda lattice and its generalizations have been the test models for various techniques and philosophies in integrable systems and wide connections are built with many other branches of mathematics. In this talk, I will talk about its connection with the so-called Hessenberg varieties among which Peterson variety and flag varieties are the most well known ones. This talk is based on a joint work with Yuji Kodama.