University of Illinois Urbana-Champaign Integrability and Representation Theory Seminar 2024

Seminar Schedule IRT Seminar

Seminar Schedule Fall 2024

Time Date Speaker Institution Location Title Abstract
12:00pm - 1:00pm October 17th, 2024 David Keating UIUC Online. Email Trung Vu (hvu at illinois dot edu) for Zoom information A Vertex Model for LLT polynomials Click for abstract
The LLT polynomials introduced by Lascoux, Leclerc, and Thibon are a family of symmetric polynomials indexed by tuples of partitions that generalize the more well-known Schur polynomials. In this talk will show how to construct a version of theses LLT polynomials as the partition function of a Yang-Baxter integrable vertex model. As a consequence, we will be able to prove that the polynomials are in fact symmetric and that they satisfy a certain Cauchy identity. Building on this we will define coupled domino tilings of the Aztec diamond and use our vertex model to enumerate them. Finally, if time permits, we will present an algorithm for determining when the LLT polynomial indexed by the pair of partitions \((\lambda^{(1)}, \lambda^{(2)})\) is equal to the LLT polynomial indexed by \((\lambda^{(2)}, \lambda^{(1)})\). The key ingredient will be the vertex model construction of the polynomials.
12:00pm - 1:00pm November 7th, 2024 Trung Vu UIUC Online. Email Trung Vu (hvu at illinois dot edu) for Zoom information Part 1: Introduction to dimer models on infinite ℤ2 periodic graphs, Kasteleyn matrix and height functions Click for abstract

The partition function is one of the key objects in statistical mechanics encoding the macroscopic behavior of the underlying models. In 1967, Kasteleyn, and independently by Fisher and Temperley, gave an explicit computation of the partition function for the dimer models on the square lattice as the determinant of the signed adjacency matrix. In this talk, I will introduce the notion of Kasteleyn matrix for the infinite, bipartite planar graphs with ℤ2-periodic property, the height function of dimer covers and how the two objects related. Understanding these objects will be crucial for constructing invariant Gibbs measure and the phase diagram of the Gibbs measures for dimers on a particular graph, represented by the amoeba. This is the first talk in a series of 3 talks on the paper "Dimer and Amoeba" by Kenyon-Okounkov-Sheffield and (if time allowed) the correspondence between amoeba and the arctic curves from the T-system dimer model.

Amoeba
12:00pm - 1:00pm November, 14th Trung Vu UIUC Online. Email Trung Vu (hvu at illinois dot edu) for Zoom information Part 2: Characteristic polynomials and spectral curves of infinite ℤ2 periodic dimer models Click for abstract

We continue the discussion from last week, extending the notion of height function and Kasteleyn theory on planar dimer models to the infinite case. The extension of the Kasteleyn matrix to the infinite planar bipartite graphs with ℤ2-periodic property allows one to define the characteristic polynomial which contains most information of the macroscopic behavior of the dimer model. If time allows, we will briefly discuss discrete complex analysis on planar bipartite graphs.
12:00pm - 1:00pm November, 21st Trung Vu UIUC Online. Email Trung Vu (hvu at illinois dot edu) for Zoom information Part 3: Amoeba and invariant Gibbs measure on infinite ℤ2 periodic dimer models Click for abstract

We finished the discussion of infinite ℤ2 periodic dimer models asymptotics by introducing the invariant Gibbs measure and the amoeba. The amoeba will act as the phase diagram of the model's asymptotics behavior
12:00pm - 1:00pm December, 5th Wonwoo Kang UIUC Online. Email Trung Vu (hvu at illinois dot edu) for Zoom information Modified Snake Graph : Type B and C Click for abstract

Fomin and Zelevinsky demonstrated that \(\theta\)-invariant triangulations of \(P_{2n+2}\) correspond bijectively to the clusters of cluster algebras of type \(B_n\) or \(C_n\). Additionally, cluster variables are associated with the orbits of the \(\theta\)-action on the diagonals of \(P_{2n+2}\). In this talk, I will present how to describe the cluster variables of type \(B_n\) and \(C_n\) with principal coefficients using the perfect matchings of modified snake graphs. This is part of ongoing work with Esther Banaian, Elizabeth Kelley, Ezgi Kantarcı Oğuz, and Emine Yıldırım.

Seminar Schedule Spring 2025

Time Date Event type Speaker Institution Location Title Abstract
4:00 pm February, 6th IRT Seminar Michael Gekhtman University of Notre Dame Altgeld Hall, Room 343 New generalized cluster structures on \(GL(n)\) - a case study Click for abstract
I will discuss a generalized cluster on \(GL(n)\) compatible with the particular Poissonbracket that is homogeneous w.r.t. two-sided action of a Poisson-Lie group \(G=GL(n) \times GL(n)\). Here the components of G are equipped with two „opposite” versions of the Cremmer-Gervais Poisson-Lie bracket. Our construction relies on birational Poisson maps that relate the Poisson homogeneous structure under investigation with the phase space of the finite Toda lattice and the Poisson dual of the Cremmer-Gervais Poisson-Lie group. This is a joint work with M. Shapiro and A. Vainshtein.
1:00 pm February, 20th IRT Seminar Leonid Petrov University of Virginia Altgeld Hall, Room 343 (for the seminar) Random Fibonacci Words Click for abstract
Fibonacci words are words of 1's and 2's, graded by the total sum of the digits. They form a differential poset (YF) which is an estranged cousin of the Young lattice powering irreducible representations of the symmetric group. We introduce families of "coherent" measures on YF depending on many parameters, which come from the theory of clone Schur functions (Okada 1994). We characterize parameter sequences ensuring positivity of the measures, and we describe the large-scale behavior of some ensembles of random Fibonacci words. The subject has connections to total positivity of tridiagonal matrices, Stieltjes moment sequences, orthogonal polynomials from the (q-)Askey scheme, and residual allocation (stick-breaking) models. Notes from talk
1:00 pm February, 27th IRT Seminar Marianna Russkikh University of Notre Dame Altgeld Hall, Room 343 Perfect t-embeddings of Hexagon Click for abstract
A new type of graph embedding called a (perfect) t-embedding, was recently introduced and used to prove the convergence of dimer model height fluctuations to a Gaussian Free Field (GFF) in a naturally associated metric, under certain technical assumptions. We will describe a construction of perfect t-embeddings for regular hexagons of the hexagonal lattice and discuss their properties. The construction provides the first example, and hence proves the existence of perfect-t-embeddings for graphs with an outer face of a degree greater than four. As a consequence, this construction leads to a new proof of GFF fluctuations for the dimer model height function on uniformly weighted hexagon.
1:00 pm April, 17th IRT Seminar Esther Banaian University of California - Riverside Altgeld Hall, Room 343 A cluster-theoretic perspective on Markov numbers and Cohn matrices Click for abstract
Markov numbers, i.e. numbers which appear in solution triples to \(x^2+y^2+z^2=3xyz\), first appeared in the context of Diophantine approximation. Cohn exhibited a connection between Markov numbers and the lengths of closed simple geodesics on the punctured torus. A byproduct is the family of Cohn matrices, which can be seen as a matrixization of Markov numbers. It is known that Markov numbers can be viewed as specializations of cluster variables in the cluster algebra from a once-punctured torus. We give a cluster-algebraic interpretation to Cohn matrices, using poset-theoretic formulas from Kantarcı-Oguz-Yıldırım and Pilaud-Reading-Schroll. We also discuss variations of the Markov equation and cluster Cohn matrices, inspired by generalized cluster algebras in the sense of Chekhov-Shapiro. This is based on joint works with Yasuaki Gyoda and with Archan Sen
1:00 pm April, 24th IRT Seminar Matthew Nicoletti University of California - Berkeley Altgeld Hall, Room 343 The Doubly Periodic Aztec Diamond Dimer Model: Gaussian Free Field and Discrete Gaussians Click for abstract
The dimer model provides a large family of random surface models whose scaling limits exhibit spatial phase separation, and have universal properties such as conformal invariance. The Aztec diamond dimer model in particular has been extensively researched due to its integrability, in order to more precisely understand the various universal behaviors of dimer models. While dimer model height fluctuations have been computed in many cases, until now the exact characterization of height fluctuations of a dimer model with gaseous facets appearing in the bulk has remained open. We analyze height fluctuations in Aztec diamond dimer models with nearly arbitrary periodic edge weights, which lead to the formation of gaseous facets in the limit shape. We show that the centered height function approximates the sum of two independent components: a Gaussian free field on the multiply connected rough region and a harmonic function with random, lattice-valued rough-gas boundary values. The boundary values are jointly distributed as a discrete Gaussian random vector. This discrete Gaussian distribution maintains a quasi-periodic dependence on N, a phenomenon also observed in multi-cut random matrix models. This is joint work with Tomas Berggren.
4:00 pm May, 1st Department Colloquium Gus Schrader Northwestern University Altgeld Hall, Room TBA Curve counts and cluster varieties Click for abstract
One can often solve counting problems depending on a discrete parameter by assembling the different counts as coefficients of a generating function, deriving a differential or algebraic equation for this generating function, and finally solving that equation. I will explain how this method can be applied to a problem of counting holomorphic maps from Riemann surfaces with boundary into $\mathbb{C}^3$, such that the boundary lands on a fixed real (Lagrangian) 3-manifold $L$. It turns out that the equations satisfied by the corresponding generating functions associated to different choices of boundary conditions $L$ can be encoded via a single algebro-geometric object known as a cluster Poisson variety. The rigid combinatorial structure of this auxiliary geometry can then be used to find an explicit formula for the solution of these equations, making manifest nontrivial properties of the original curve counts. Based on joint works with Mingyuan Hu, Linhui Shen, and Eric Zaslow.

Organizers of the IRT seminar are Philippe Di Francesco, Rinat Kedem, David Keating, Wonwoo Kang and Trung Vu