## KU Combinatorics Seminar Spring 2019

Friday 1/25
Pam Harris (Williams College)
Kostant's Partition Function

Abstract: In this talk we introduce Kostant's partition function which counts the number of ways to represent a particular weight (vector) as a nonnegative integral sum of positive roots of a Lie algebra. We provide two fundamental uses for this function. The first is associated with the computation of weight multiplicities in finite-dimensional irreducible representations of classical Lie algebras and the second is in the computation of volumes of flow polytopes. We provide some recent results in the representation theory setting and state a direction of ongoing research related to the computation of the volume of a new flow polytope associated to a Caracol diagram.

Friday 2/1
Organizational meeting

Friday 2/8
Jeremy Martin
Report from GRWC 2018: Parking Functions and Elser's Invariant

Abstract: I will report on a project from the 2018 Graduate Research Workshop in Combinatorics, about enumeration for completions of parking functions and potential consequences for Pitman-Stanley polytopes.

Friday 2/15
Seminar cancelled due to inclement weather

Friday 2/22
Kevin Marshall
What Is A Modulus On Graphs?

Abstract: We discuss the notion of modulus on graphs as first developed by Nathan Albin et al. in, e.g., arXiv:1401.7640 and arXiv:1805.10112. The concept of modulus has its origins in analysis dealing with families of curves. Modulus on graphs can be thought of as the discrete analogue to this idea. We will cover some basic facts about modulus on graphs and in particular focus on using modulus with walks and trees of a graph.

Friday 3/1
Mark Denker
What is Polya's Theorem?

Friday 3/8
Federico Castillo
Deformations of Coxeter permutahedra and Coxeter submodular functions

Abstract: One way to decompose a polytope is to represent it as a Minkowski sum of two other polytopes. These smaller pieces are naturally called summands. Starting from a polytope we want to explore the set of all summands. This set can be parametrized by a polyhedral cone, called deformation cone, in a suitable real vector space. We focus on the case where the starting polytope is a Coxeter permutahedron, which is a polytope naturally associated with a root system. This generalizes the type A case, which corresponds to generalized permutohedra. This is joint work with Federico Ardila, Chris Eur, and Alexander Postnikov.

Friday 3/15
No seminar (Spring Break)

Friday 3/22
Justin Lyle
Rank Selection and Depth Conditions for Balanced Simplicial Complexes

Abstract: A simplicial complex is said to be balanced if its vertices can be colored so that none of its faces contain more than one vertex of a given color. We focus on rank selected subcomplexes of balanced simplicial complexes, which are formed by excluding all vertices of a given set of colors. In particular, we are interested in the depth properties of these subcomplexes and how they connect with those of the original complex. By passing to a barycentric subdivision, we discuss how our results can be applied to any simplicial complex. This is joint work with Brent Holmes.

Friday 3/29
TBA

Friday 4/5
Carl Lee (University of Kentucky)
The a,b,c,d,C,D,f,g,h of Polytopes

Abstract: I will sample some different methods in the enumeration of chains of faces of convex polytopes, and discuss their relationships.

Friday 4/12
Independence Complexes of Finite Groups

Abstract: Understanding generating sets for groups is of interest in group theory, as certain problems involving groups are made easier by appropriate choices of generators. Our goal is to describe minimal generating sets for certain finite groups in a visual way. We will explore some interesting combinatorics that arises from this view. More specifically, let $$G$$ be a finite group. We define an independent set of $$G$$ to be a collection of group elements that is a minimal generating set for some subgroup of $$G$$. These independent sets form a simplicial complex (called the Independence Complex) whose vertices are elements of $$G$$ and whose faces of size $$k$$ correspond to independent sets with $$k$$ generators. We describe the structure and combinatorics of the resulting simplicial complex.

Friday 4/19
Alex McDonough (Brown University)
Chip-Firing on Simplicial Complexes and Matroids

Abstract: The sandpile group and its relation to spanning trees has been studied extensively in the graphical setting in the context of chip-firing. A more recent area of research has been to generalize chip-firing results to a larger class of objects. In this talk, we explore chip-firing on simplicial complexes as well as regular matroids. Through combining these topics, we produce a bijective proof for Duval-Klivans-Martin's Simplicial Matrix Tree Theorem on a particular class of simplicial complexes. This bijection provides representatives for the elements of the sandpile groups for these complexes that depend only on a vertex ordering. This talk does not assume any prior knowledge of chip-firing or matroids.

Friday 4/26
Souvik Dey
Generalized Flag Varieties: An Introduction

Abstract: Classically, a full flag in a vector space of dimension n is defined to be an increasing sequence of vector subspaces of length n, starting with dimension 0 and with increment 1. The collection of all such (full) flags, called the full flag variety, can be viewed as a Zariski closed subset of a product of Grassmaninan varieties, from which the collection obtains a variety structure, so that it is called a full flag variety. It can be seen that the general linear group acts transitively on this variety, the invariant subgroup of the action being the group of upper-triangular matrices, also called the Borel subgroup of the general linear group, so that the variety can be parametrized by the co-sets of this Borel subgroup. All these concepts can actually be generalized in the context of algebraic groups, where one can define a notion of Borel subgroups, and the homogeneous space of that is called a generalized full (sometimes also called complete) flag variety. And in general, a generalized flag-variety is the quotient space of a parabolic subgroup. In this talk, we will try to make sense of these concepts of Borel subgroups, parabolic subgroups and flag varieties in the general context of algebraic groups and mention in what way they are special and try to give some results explaining their geometry.

Friday 5/3
McCabe Olsen (Ohio State)
Birkhoff polytopes of different type and the orthant-lattice property

Abstract: Given a $$d$$-dimensional lattice polytope $$P$$, we say that $$P$$ has the orthant-lattice property (OLP) if the subpolytope obtained by restriction to any orthant is a lattice polytope. While this property feels somewhat contrived, it can actually be quite useful in verification of discrete geometric properties of $$P$$. In this talk, we will discuss a number of results for the existence of triangulation and the integer decomposition property for reflexive OLP polytopes. One such polytope which fits into the program is a type-B analogue of the Birkhoff polytope and its dual polytope, the investigation of which led to interest in this property.

Friday 5/10
No seminar (Stop Day)

Here's our potential list of speakers and "What is..." topics (left over from last fall, and subject to change):

• Dylan - universal algebra
• Jeremy - root systems, $$E_8$$ (want the full story? Take Math 996!)
• Ken - Cayley graphs
• Kevin - Borel-fixed ideals
• Mark - Polya's theorem

For seminars from previous semesters, please see the KU Combinatorics Group page.

Last updated Wed 4/24/19