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CombinaTexas: Combinatorics in the South-Central U.S.
April 19--20, 2008
University of Texas at El Paso

Speaker: Marcelo Aguiar

Title updated: Products of symmetric functions

Abstract updated: We introduce a product on the space of symmetric functions that interpolates between the classical "internal" and "external" products (which are constructed in terms of tensor products and induction of representations). We call it the "Heisenberg" product.

The Heisenberg product exists as well on the space of non-commutative symmetric functions. At this level it interpolates between Solomon's product (the "descent algebra") and the usual product of non-commutative symmetric functions (as defined by Thibon et al).

The Heisenberg product of symmetric functions is best understood in terms of "species". On the space of non-commutative symmetric functions and on certain larger spaces, the product is best understood in terms of certain Hopf algebraic constructions.

We will discuss these algebraic notions and the combinatorics underlying it.

This is joint work with Walter Ferrer and Walter Moreira.

Speaker: Maria Axenovich

Title: Induced subgraphs: avoiding, reconstructing, coloring

Abstract: For a graph G=(V, E), graph G'=(V', E') is an induced subgraph of G if
V' \subseteq V, and for any x,y \in V', {x,y} \in E' iff {x,y} \in E. In other words, a G' is an induced subgraph of G if it can be obtained from G by vertex deletions.

The study of induced subgraphs not only continues being a very challenging and exciting area of extremal graph theory, but also becomes very attractive for applications such as ones in biological networks. In particular, compared to ordinary subgraphs, induced subgraph carry information not only about existing connections but also about missing connections between the nodes of the network.

In this talk I will address several problems and new results in the following areas:

  1. Avoiding fixed induced subgraph via edge-deletions and edge-additions, i.e., so-called graph editing.
  2. Identifying graph structure from the limited information on induced subgraphs - topic closely related to graph reconstruction.
  3. Unavoidable induced subgraphs in graph partitions, i.e. Ramsey and anti-Ramsey problems.

Speaker: Patricia Hersh

Title: Regular cell complexes in total positivity

Abstract: We give a new criterion for determining whether a finite CW complex is regular. This involves both combinatorial conditions on the closure poset and also topological conditions on the codimension one cell incidences. We will also discuss how this applies to a conjecture of Fomin and Shapiro that certain stratified totally non-negative spaces with the Bruhat intervals as their closure posets are regular CW complexes. We will not assume familiarity with CW complexes.

Speaker: Nathan Reading

Title: Cluster algebras and infinite associahedra

Abstract: Cluster algebras of finite type are relatively well understood. Each has as its combinatorial foundation a simplicial polytope known as a generalized associahedron. (A special case is the usual associahedron, whose vertices are triangulations of a convex polygon, with edges corresponding to diagonal flips.) The generalized associahedra are described combinatorially in terms of clusters of roots in a finite root system. Recently David Speyer and I gave another construction of cluster complexes as simplicial fans which coarsen the fan cut out by the reflecting hyperplanes of a finite Coxeter group W. This construction is deeply rooted in the combinatorics of W (in the form of "sortable elements") and in the lattice theory of the weak order on W (in the form of "Cambrian lattices").

In this talk, I will review the finite-type results, then discuss joint work in progress with David Speyer to move beyond finite type. The combinatorics and geometry of sortable elements in an infinite Coxeter group reveal a tantalizing glimpse of "infinite associahedra." Specifically, we construct fans (conjecturally) isomorphic to subfans of cluster complexes of infinite type. I will also discuss connections to noncrossing partitions.

I will assume no prior knowledge of Coxeter groups, cluster algebras, associahedra or noncrossing partitions.

Speaker: Anne Shepler

Title: Reflection Groups and Invariant Theory in Nonzero Characteristic

Abstract: The classical theory of reflection groups over the real and complex numbers began with geometric investigations. Coxeter, Chevalley, Shephard and Todd, and Solomon (among others) showed that these groups enjoy a unique and captivating invariant theory by interpretating their geometry. What happens when we work over finite fields, or fields of nonzero characteristic in general? Classical results fail and the role played by the geometry of roots and reflecting hyperplanes becomes perplexing. We discuss how geometry interacts with invariant theory in the modular setting.

Speaker: Lauren Williams

Title: Permutation tableaux in geometry, statistical physics, and combinatorial Hopf algebras

Abstract: We will explain how the combinatorics of permutation tableaux appears in three seemingly different settings. Permutation tableaux are a distinguished subset of Alex Postnikov's "Le-diagrams," which index cells in the totally non-negative part of the Grassmannian. In joint work with Sylvie Corteel, we have shown that one can compute the steady state probability of each state in the asymmetric exclusion process (a model from statistical physics involving particles hopping on a one-dimensional lattice) by counting permutation tableaux according to certain statistics. And in joint work with Jean-Christophe Novelli and Jean-Yves Thibon, we have introduced some new non-commutative Hall-Littlewood polynomials: when we express them in terms of a certain basis, we see these same steady state probabilities appearing in the transition matrix.

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Copyright ©2008 by Art Duval
Last Modified on Apr. 18, 2008.