| Amol Aggarwal (Columbia University) | Half-stationary vertex models and fusion. |
| TBA. | |
| Ben Brubaker (University of Minnesota) | Integrable systems and p-adic representation theory |
| We'll survey recent results on using lattice models to represent special functions from p-adic representation theory, which include and generalize a number of important examples from symmetric function theory. We emphasize the use of integrability in demonstrating various properties of and identities for the resulting partition functions. A common theme is that all boundary conditions on our finite lattice models have an associated meaning for p-adic representation theory. Our work is joint with Buciumas, Bump, and Gustafsson, and we'll discuss relations with work of various combinations of Borodin, Gorbounov, Korff, Wheeler, and Zinn-Justin motivated by integrable probability and Schubert calculus. | |
| Iva Halacheva (Northeastern University) | Branching in Schubert calculus |
| A classical question in Schubert calculus is the study of the pullback in (equivariant) cohomology of the diagonal inclusion of a type A Grassmannian Gr(k,n), namely the multiplication of Schubert classes. This was addressed via puzzles by Knutson—Tao, and more recently reinterpreted via quantum integrable systems and generalized to higher-step partial flag varieties by Wheeler—Zinn-Justin and Knutson—Zinn-Justin. A similarly natural question is understanding, via the Schubert bases, the pullback of the inclusion of a symplectic Grassmannian SpGr(k,2n) into Gr(k,2n). We address this question via puzzles and the 5-vertex model, and generalize to cotangent bundles and the 6-vertex model. The Schubert classes are then replaced with Maulik—Okounkov classes, in the setup of Lagrangian correspondences between symplectic resolutions. This is joint work in progress with Allen Knutson and Paul Zinn-Justin. | |
| Jules Lamers (University of Melbourne) | How coordinate Bethe ansatz works for Inozemtsev model |
| Three decades ago, Inozemtsev discovered an isotropic long-range spin chain with elliptic pair potential that interpolates between the Heisenberg and Haldane--Shastry (HS) spin chains while admitting an exact solution throughout, based on a connection with the elliptic quantum Calogero--Sutherland (eCS) model. Though Inozemtsev's spin chain is widely believed to be quantum integrable, the underlying algebraic reason for its exact solvability is not yet well understood. As a step in this direction we refine Inozemtsev's `extended coordinate Bethe ansatz' and clarify various aspects of the model's exact spectrum and its limits.
We identify quasimomenta in terms of which the M-particle energy is close to being additive, as one would expect from the limiting models; our expression is additive iff the eCS energy is so. This moreover allows us to rewrite the M-particle energy and Bethe-ansatz equations on the elliptic curve, turning the spectral problem into a rational problem as might be expected for an isotropic spin chain. We treat the M=2 particle sector and its limits in detail. We identify an S-matrix that is independent of positions despite the more complicated form of the extended coordinate Bethe ansatz. We show that, as the interpolation parameter varies, the `scattering states' from Heisenberg become Yangian highest-weight states for HS, while bound states become (sl2-highest weight versions of) affine descendants of the magnons from M=1. For bound states we find an equation that, for given Bethe integers, relate the `critical' values of the spin-chain length and the interpolation parameter for which the two complex quasimomenta collide; it reduces to the known equation for the `critical length' in the Heisenberg limit. This talk is based on joint work with Rob Klabbers (Nordita, Stockholm). |
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| Alexander Molev (University of Sydney) | Symmetrization map, Casimir elements and Sugawara operators |
| The canonical symmetrization map is a g-module isomorphism between the symmetric algebra S(g) of a finite-dimensional Lie algebra g and its universal enveloping algebra U(g). This implies that the images of g-invariants in S(g) are Casimir elements. For each simple Lie algebra g of classical type we consider basic g-invariants arising from the characteristic polynomial of the matrix of generators. We calculate the Harish-Chandra images of the corresponding Casimir elements. By using counterparts of the symmetric algebra invariants for the associated affine Kac-Moody algebras, we obtain new formulas for generators of the centers of the affine vertex algebras at the critical level. Their Harish-Chandra images are elements of classical W-algebras which we produce in an explicit form. | |
| Leonid Petrov (University of Virginia) | Symmetric functions from vertex models |
| I will discuss how properties of symmetric functions (such as Schur and Hall-Littlewood functions and their generalizations) arise from studying integrable vertex models. The focus will be on (1) summation identities; (2) a new class of continuously-indexed symmetric functions generalizing the 2F1 hypergeometric functions which we call the spin-Whittaker functions. The second part is based on a joint work with Matteo Mucciconi. | |
| Anna Puskás (University of Queensland) | A correction factor for Kac-Moody groups and t-deformed root multiplicities |
|
The study of infinite dimensional analogues of classical formulas from
the theory of p-adic groups gives rise to a certain correction factor.
For example, Macdonald's formula for the spherical function and the
Casselman-Shalika formula, when extended to the affine, and general
Kac-Moody setting, all have this feature. Its presence captures the
ambiguity remaining after considering functional equations corresponding
to elements of an infinite Weyl group.
We will discuss this correction factor. In affine type, it is known by Cherednik's work on Macdonald's constant term conjecture. More generally, it can be represented as a collection of polynomials of t indexed by positive imaginary roots; these are deformations of root multiplicities. Methods of computing imaginary root multiplicities, such as the Peterson algorithm and the Berman-Moody formula can be generalized to compute the correction factor for any t. They both reveal some properties of the correction factor and raise further questions and conjectures about its structure. This is joint work with Dinakar Muthiah and Ian Whitehead. |
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| Tomohiro Sasamoto (Tokyo Institute of Technology) | Current moment formulas for 1D exclusion processes |
| We give a proof of a formula for the moments of the integrated current for the one-dimensional symmetric simple exclusion process (SEP) with double-sided Bernoulli initial condition. The formula itself had been found in our previous paper and it was the basis for studying the large deviation of a tagged particle position. We obtain the moment formula for SEP by taking a symmetric limit of a deformed moment formula for asymmetric simple exclusion process (ASEP). We explain that an intricacy of the symmetric limit can be handled by a novel recursion relation satisfied by a part of the integrand of the deformed moment formula. | |
| Travis Scrimshaw (University of Queensland) | Refined dual Grothendieck polynomials from integrability |
| The (symmetric) Grothendieck polynomials, which are the generating function of set-valued tableaux and arose from Schubert calculus, form a basis for symmetric functions. The dual Grothendieck polynomials are the corresponding dual basis under the standard inner product where the Schur functions are orthonormal. A refined version of a dual Grothendieck polynomial was introduced by Galashin, Grinberg, and Liu. In this talk, we will construct a solvable lattice model whose partition functions are refined dual Grothendieck polynomials by using a combinatorial interpretation due to Lam and Pylyavskyy and the classical nonintersecting lattice path description of Schur functions. Using our lattice model, we obtain a Jacobi-Trudi formula, Cauchy-type identity, a new symmetry, and refined versions of previous formulas. We can also deform our model to use the classical five-vertex model for Schur functions, which then allows us to add an additional natural parameter. This is joint work with Kohei Motegi. | |
| Junichi Shiraishi (University of Tokyo) | Some conjectures concerning non-stationary Ruijsenaars functions |
| Abstract. | |
| Michael Wheeler (University of Melbourne) | Puzzles for non-symmetric Schur functions |
| TBA. | |
| Paul Zinn–Justin (University of Melbourne) | Bosonic lattice models and honeycombs for Grothendieck polynomials |
| After a brief review of exactly solvable models associated to Grothendieck polynomials, I will discuss some recent work in collaboration with A. Gunna, in which we obtain new representations of Grothendieck polynomials and their duals (in the sense of product/coproduct duality) as partition functions of "bosonic" lattice models. This leads to product rules (generalizing the Littlewood-Richardson rule) in terms of certain modified honeycombs. | |