Independence polynomials are generating functions for the number of independent sets of each cardinality in a graph G. In addition to encoding useful information about the graph (such as the number of vertices, the number of edges and the independence number), the analytic and algebraic properties can say much about the shape and inter-dependence of…

Chapoton triangles are polynomials in two variables defined by Coxeter-Catalan objects. These polynomials are related by some remarkable identities that only depend on the rank of the associated (finite) Coxeter system. The multidimensional Catalan numbers enumerate the number of standard Young tableaux of a rectangular shape. In this talk I will introduce some other combinatorial objects counted by these numbers. I…

In this talk, we will explain a (mysterious?) connection between the combinatorics of affine buildings and representation theory in type A. If a group acts simply and transitively on the vertices of an affine building in type A_n, it gives rise to a certain combinatorial structure called a triangle presentation. We will describe how triangle presentations…

We consider a finite group scheme G, and its associated representation category rep G.  Here one can think of a finite discrete group, or an infinitesimal group scheme, such as the kernel of the r-th Frobenius map for GL_n over F_p.  Via a standard tensor categorical construction one has Drinfeld's center Z(rep G) of the…

If you take a simple finite-dimensional Lie algebra g and tensor it with the Laurent polynomials in one variable, then you will get an infinite-dimensional Lie algebra known as a loop algebra. Affine Lie algebras are the central extensions of such loop algebras and their representations have been of interest to several mathematicians. What happens if we tensor g with…

Modular tensor categories arise naturally in many areas of mathematics, such as conformal field theory, quantum groups and Hopf algebras, low dimensional topology, representations of braid groups, and they have important applications in condensed matter physics, modeling topological phases of matter. In this talk, I will start by introducing the relevant concepts (modular, braided and…

Linear extensions of posets are important objects in enumerative and algebraic combinatorics that are difficult to count in general. Families of posets like straight shapes and $d$-complete posets have hook-length product formulas to count linear extensions, whereas families like skew shapes have determinant or positive sum formulas like the Naruse hook length formula from 2014.…

For this week, we have a special Algebra and Combinatorics meet & greet where we'll get to chat informally and discuss some ideas for the seminar and the upcoming talks.   Join Zoom Meeting https://ncsu.zoom.us/j/98353887156?pwd=cUN6VnNwbHQ5Vyt2aFVCZmVPNm5nQT09 Meeting ID: 983 5388 7156 Passcode: Alg&Com21

In this talk, I will share with you what kind of problems I work on and what's my motivation. We will talk about the representation theory of finite groups and symmetric functions, and how algebraic combinatorics appears in the less expected places. This talk aims to be mostly informal and accessible for grad students.  Speaker’s…

Fusion categories are algebraic structures that generalize the representation categories of finite groups. I will explain how fusion categories have become involved in diverse areas of mathematics and physics, from topologically ordered phases of matter in 2-dimensions to quantum symmetries of noncommutative spaces.   Jointly in person and virtually on Zoom. SAS 4201 for in-person…

I will discuss some algebraic aspects of recent work with Raphael Rouquier on a tensor product operation for categorified representations of U_q(gl(1|1)^+) and its connections to Heegaard Floer homology. Speaker’s webpage: https://sites.google.com/usc.edu/manion/home Jointly in person and virtually on Zoom. SAS 4201 for in-person participation. The Zoom link is sent out to the Algebra and Combinatorics…

First, I will make a general introduction to vertex algebras. Then, I will mention some results of recent work with Bojko Bakalov on Logarithmic vertex algebras.   Jointly in person in SAS 4201 or virtually on Zoom. The Zoom link is sent out to the Algebra and Combinatorics mailing list, please contact Corey Jones at cmjones6@ncsu.edu to…

I will tell you about my dissertation work on two variants of stable Grothendieck polynomials and their combinatorics. Relevant combinatorial objects include crystals (edge-labelled directed digraphs from representation theory), tableaux (numbers in boxes with rules), decreasing factorizations (numbers in parentheses), and insertion algorithms (how to put numbers in boxes). Background in algebraic combinatorics is helpful…

In a non-local game, two non-communicating players cooperate to convince a referee about a strategy that does not violate the rules of the game. A quantum strategy for such a game enable players to determine their answers by performing joint measurements on a shared entangled state. In this talk we will concentrate on non-local games…

The totally positive flag variety is the subset of the complete flag variety Fl(n) where all Plücker coordinates are positive. By viewing a complete flag as a sequence of subspaces of polynomials of degree at most n-1, we can associate a sequence of Wronskian polynomials to it. I will present a new characterization of the…

We establish the conjecture of Reiner and Yong for an explicit combinatorial formula for the expansion of a Grothendieck polynomial into the basis of Lascoux polynomials. This expansion is a subtle refinement of its symmetric function version due to Buch, Kresch, Shimozono, Tamvakis, and Yong, which gives the expansion of stable Grothendieck polynomials indexed by…

It is a classical result that the simple algebras in the category of finite dimensional vector spaces are precisely the n x n matrix algebras. The notion of algebras in more general tensor categories is easy to formulate, and we can ask for classification results in these categories. Such classification results have broad applications to conformal field…

The theory of P-partitions was developed by Stanley to understand/solve several enumerations problems and representations theory problems. Together with the work of Gessel, this led to the development of the space of quasisymmetric functions. Schur functions are naturally understood in the world of quasisymmetric functions as a sum over standard tableaux of Gessel fundamental functions.…

Jointly in person and virtually on Zoom. SAS 4201 for in-person participation. The Zoom link is sent out to the Algebra and Combinatorics mailing list, please contact Corey Jones at cmjones6@ncsu.edu to be added. Abstract: The Newell-Littlewood numbers are defined in terms of the Littlewood-Richardson coefficients. Both arise as tensor product multiplicities for a classical Lie group. A.…

Coxeter groups were famously proven to be automatic by Brink and Howlett in 1993 and the automaticity of these groups has been an area of continued interest since. In this talk, we give a brief history and summary of recent developments in this area, and we introduce the theory of Regular Partitions of Coxeter groups.…