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
1:00 pm - 1:50 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 - 1:50 pm February, 20th IRT Seminar Leonid Petrov University of Virginia Altgeld Hall, Room 343 (for the seminar) TBA Click for abstract
TBA
1:00 pm February, 27th IRT Seminar Marianna Russkikh University of Notre Dame Altgeld Hall, Room 343 TBA Click for abstract
TBA

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