Title: Kazhdan-Lusztig polynomials of thagomizer graphs Abstract: To a graph G, one can associate a polynomial with non-negative integer coefficients called the Kazhdan-Lusztig polynomial of G. More generally, you can obtain the Kazhdan-Lusztig polynomial of any matroid, but today we will focus on the specialization to graphs. The Kazhdan-Lusztig theory for matroids was developed in analogy with the classical theory for Coxeter groups, though there are some important differences which I will touch on lightly. In this talk, we will construct the defining recursion for the Kazhdan-Lusztig polynomial of thagomizer graphs and use this obtain a closed form for the coefficients of the polynomial. No prior knowledge of matroids or Kazhdan-Lusztig polynomials will be assumed.

Title: “Cold atoms, SU(N) symmetry and Young tableaux”



Abstract:  Symmetries play a foundational role in our understanding of physics.

 It is often the case that unexpected symmetries can emerge from unexpected places.

I will discuss a particular example in systems of cold atoms that realize SU(N) symmetry,

where N can be as large as ~10.  I will then describe how these systems can be treated

numerically by using results on the symmetric group dating back to Young.

Discrete CATS Seminar

2pm, Monday Feb 13

POT 745

Speaker: Laura Escobar, UIUC

Discrete CATS Seminar

 

Title: Universal partial words

 

Abstract: A universal word for an alphabet A and a positive integer n is a word containing each of the words of length n over A as a substring exactly once. For instance, de Bruijn sequences are (cyclic) universal words. Universal partial words, introduced by Chen, Kitaev, Mutze, and Sun in 2016, allow for a wild-card character , which can stand for any letter in the alphabet. We settle and strengthen conjectures posed in the same paper where this notion was introduced. For non-binary alphabets, we show that universal partial words have periodic structure and are cyclic, and we give number-theoretic conditions on the existence of universal partial words. In addition, we provide an explicit construction for a family of universal partial words over alphabets of even size. This talk is based on joint work with Bennet Goeckner, Corbin Groothius, Brian Kell, Pamela Kirkpatrick, Rachel Kirsch, and Ryan Solava.

 

Title: P-partitions Revisited

Abstract: We will be discussing the paper P-partitions Revisited, which attempts, for a given poset P, to connect algebraic invariants of the ring of P-partitions to the structure of P itself, i.e., gives a partial answer to the question: What poset properties survive the process of "algebrizing", and in what form?  We will introduce the relevant objects (Linear extensions, P-partition rings, Hilbert Series), and give sketches of their constructions and the relationships between them. 

Discrete CATS Seminar

Monday, October 31, 2016

2PM, POT 745

Speaker: Federico Ardila, San Francisco State University

Title: The algebraic and combinatorial structure of generalized permutahedra

Abstract: Generalized permutahedra are a beautiful family of polytopes which are known to have a rich combinatorial structure. We explore the Hopf algebraic structure of this family, and use it to unify old results, prove new results, and answer open questions about families of interest such as graphs, matroids, posets, trees, set partitions, hypergraphs, and simplicial complexes. As a corollary, we obtain a new polyhedral approach for computing the multiplicative and compositional inverses of a power series.  The talk is based on joint work with Marcelo Aguiar, and will assume no previous knowledge of Hopf algebras or generalized permutahedra.

Title: Properties of s-lecture hall polytopes

Abstract: Arising from the well-studied s-lecture hall partitions, we will discuss the s-lecture hall polytopes. In recent work, T. Hibi, A. Tsuchiya, and I were able to characterize algebraic and geometric properties for  s-lecture lecture hall polytopes, such as the Fano, reflexive, and Gorenstein properties as well as the integer decomposition property (IDP), for certain classes of s-sequences. We also make several conjectures. At the end of the talk, we will briefly discuss related polytopes, namely the partition polytope and the Gelfand–Tsetlin polytope, and recent work of P. Alexandersson which suggests that our conjectures may not be true.

 

Title: Excedance Algebra and box polynomials

Abstract: In this talk we will introduce the excedance algebra and a related sequence of polynomials known as the box polynomials. We will prove recurrence relations for the polynomials and characterize their roots. We will end the talk by showing the box polynomials can also be defined in terms of delta operators.

 

Title:  Independence Complexes, Matching Trees, and Discrete Morse Theory

Abstract:  An independent set in a combinatorial graph is a subset of vertices that are pairwise non-adjacent.  Since set independence is preserved under deletion of elements, we can construct a simplicial complex from the independent sets of a graph of interest.  In this talk, we will survey some of the tools from discrete Morse theory that are useful for analyzing these so-called independence complexes.  We will also discuss some recent results pertaining to the independence complex of small grid graphs including, but not limited to, a particular algorithm giving cell-counting recursions that connect back to some interesting combinatorial sequences.

Title: Ehrhart polynomials with negative coefficients



Abstract: The Ehrhart polynomials of integral convex polytopes count integer points under dilations of the polytopes. In this talk, I will discuss the possible sign patterns of the coefficients of Ehrhart polynomials of integral convex polytopes. While the leading terms, the second leading terms and the constant of Ehrhart polynomials are always positive, the other terms aren't necessarily positive.  In fact, some examples of Ehrhart polynomials with negative coefficients were known before. For arbitrary dimension, I will describe a construction of Ehrhart polynomials with negative coefficients.