Time | Date | Speaker | Institution | Location | Title | Abstract |
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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. ![]() |
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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. |
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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 |
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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. |
Time | Date | Event type | Speaker | Institution | Location | Title | Abstract |
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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. | |||||||
1:00 pm | February, 27th | IRT Seminar | Marianna Russkikh | University of Notre Dame | Altgeld Hall, Room 343 | TBA | Click for abstract |
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1:00 pm | April, 24th | IRT Seminar | Matthew Nicoletti | University of California Berkeley | Altgeld Hall, Room 343 | TBA | Click for abstract |
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1:00 pm | May, 1st | IRT Seminar | Gus Schrader | Northwestern University | 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