In my talk I will discuss some remarkable correspondence between symplectic varieties and vertex algebras, which has been discovered by physicists in the study of the four-dimensional N = 2 superconformal field theories. In the special class of the four-dimensional N = 2 superconformal field theories called the “theory of class S”, such correspondence is mathematically explained in terms of the…
Many questions in combinatorics, probability and thermodynamics can be reduced to counting lattice paths (walks) in regions of the plane. A standard approach to counting problems is to consider properties of the associated generating function. These functions have long been well understood for walks in the full plane and in a half plane. Recently much attention has focused on walks…
Classical Schur Q-functions describe characters of a queer Lie superalgebra, projective representations of a symmetric group and provide solutions of a BKP hierarchy. This talk is devoted to properties of a generalization of Schur Q-functions - factorial Q-functions, including a particular important case of shifted Schur Q-functions.
We consider the action of the group SL_n x SL_n on the space of m-tuples of n x n matrices by simultaneous left-right multiplication. Visu Makam and the speaker recently proved that invariants of degree at most mn^4 generate the invariant ring. This result has interesting applications in algebraic complexity theory and is related to the notion of non-commutative rank.…
Scattering diagrams arose in the algebraic-geometric theory of mirror symmetry. Recently, Gross, Hacking, Keel, and Kontsevich applied scattering diagrams to prove many longstanding conjectures about cluster algebras. Scattering diagrams are certain collections of codimension-1 cones, each weighted with a formal power series. In this talk, I will introduce cluster scattering diagrams and their connection to cluster algebras, focusing on rank-2 (i.e.…
The flow polytope associated to an acyclic graph is the set of all nonnegative flows on the edges of the graph with a fixed netflow at each vertex. We will examine flow polytopes arising from permutation matrices, alternating sign matrices and Tesler matrices. Our inspiration is the Chan-Robins-Yuen polytope (a face of the polytope of…
Cyclic descents of a permutation were defined by Cellini, by allowing the permutation to wrap around as if the last entry was followed by the first. A natural question is whether a similar, well-behaved notion of cyclic descents exists for standard Young tableaux (SYT). We conjectured that such a notion exists for SYT of any…
Keller introduced the concept of maximal green sequences. Brustle-Dupont-Perotin proved that tame quivers have finitely many maximal green sequences. We have generalized the result to m-maximal green sequences. This talk will include a gentle introduction to tame path algebras, their indecomposable modules, silting objects and their mutations, the Auslander-Reiten quiver of bounded derived categories of tame path algebras and the outline…
Given a graph G on n vertices, Postnikov defined a graph associahedron P_G as an example of a generalized permutohedron, a polytope whose normal fan coarsens the braid arrangement. Combinatorially, each face of P_G corresponds to certain collections of compatible subgraphs of G called tubings. Graph associahedra were introduced independently by Carr and Devadoss and…
The number of inversions of a permutation is the number of pairs (i < j) for which w_i > w_j. This important statistic that arises in many contexts, including as the minimum number of simple transpositions needed to express the permutation and, equivalently, as the rank function for weak Bruhat order on the symmetric group.…
Following Beilinson and Drinfeld, we describe vertex algebras as Lie algebras in a certain pseudo-tensor category. More precisely, starting from a vector superspace V with a linear operator on it, we introduce the operad of n-ary chiral operations on V. When V is equipped with a vertex algebra structure, this allows us to define the cohomology of V with…
The finite minuscule and d-complete posets generalize Young and shifted Young diagrams. They have many nice combinatorial properties; for example, minuscule posets are Gaussian and Sperner, and d-complete posets have the hook length and jeu de taquin properties. Infinite analogs of the colored minuscule posets were used by R.M. Green to construct representations of many…
Multiline queues were introduced by Ferrari and Matrin as a tool for understanding the steady state of the Totally Asymmetric Simple Exclusion Process (TASEP) on a ring. Since then, they have attracted independent interest as combinatorial objects. A queue can be described as a transformation of words by a combinatorial rule (related to the Lascoux-Schützenberger action of the symmetric group). A…
A Kirillov-Reshetikhin(KR) module is a certain finite dimensional U_q'(\mathfrak{g}) module that is determined by its Drinfeld polynomials. KR modules are an important class of modules for quantum groups with many applications to Mathematical physics. KR modules are conjectured to have many nice properties, one of which is the existence of a crystal basis. In this talk, we will give a…
Yangians are one of the main examples of quantum groups introduced by Drinfeld and have found applications in combinatorics, representation theory and algebraic geometry. It is well-known that the R-matrix presentation of the Yangian in type A yields generators of its Drinfeld presentation. It has been an open problem to extend this result to the remaining types since…
The pancake graph has the elements of the symmetric group as vertices and there is an edge between two permutations if there is a prefix reversal that transforms one permutation into the other. One can similarly define the burnt pancake graph using signed permutations instead of permutations. Since these graphs are Cayley graphs, they have several interesting properties such as being regular and…
Kirillov-Reshetikhin (KR) modules are a special class of finite-dimensional modules for affine Lie algebras that have deep connections with mathematical physics. One important aspect is that they are conjectured to have crystal bases, which is known except for affine type E and F (and its dual). One of the open problems in KR crystals is…
In their celebrated 1993 paper, Brink and Howlett proved that all finitely generated Coxeter groups are automatic. In particular, they constructed a finite state automaton recognising the language of reduced words in a Coxeter group. This automaton is not minimal in general, and recently Christophe Hohlweg, Philippe Nadeau and Nathan Williams stated a conjectural criteria…
Given a positive-definite even lattice Q, one can construct a lattice vertex algebra V. An important problem in vertex algebra theory and conformal field theory is to classify the representations of the subalgebra of fixed points, known as an orbifold, corresponding to an isometry of the underlying lattice. Once the representations are known, one can calculate their characters to further…
An important construction for (the information theoretic version of) semantic security is a "Biregular Irreducible Function" (BRI). These can be constructed from a complete biregular graph on $2^k d \times 2^k d$ by by coloring it with $2^k$ colors in such a way that each vertex has degree $d$ in each color. Good BRI's are…