A long-standing conjecture of Stanley and Stembridge asks whether certain symmetric function invariants of graphs are “e-positive,” i.e. expand positively in the basis of elementary symmetric functions. Efforts to resolve Stanley—Stembridge have raised the wider question of whether other symmetric functions are e-positive, particularly for symmetric invariants of other interesting objects. This talk will describe an instance of e-positivity arising in the study of modules of the unipotent upper triangular groups over a finite field. I use a collection of novel module homomorphisms to prove e-positivity, and these maps implicate a well-known algorithm from the study of Iwahori—Hecke algebras. Throughout, I will connect these results to previously known connections between the unipotent upper triangular group and symmetric functions in non commuting variables.

Positivity in unipotent symmetric function invariants and Hecke traces Sponsored by the Meyer Fund

Tue, May. 7 12:10pm (MATH …

Kempner

Luis David Garcia Puente (Colorado College)

X

Let G be a finite, connected graph. An arithmetical structure on G is an assignment of positive integers to the vertices such that, at each vertex, its label divides the sum of the labels at adjacent vertices (counted w/ multiplicity). Arithmetical structures naturally generalize the notion of the Laplacian matrix of a graph. They also appear in the context of arithmetical geometry.

In this talk, I will present a general introduction to the topic, we will focus on the combinatorics of arithmetical structures on certain graph families including some counting results that relate arithmetical structures on a path graph and Catalan numbers. We will also discuss some results about the associated finite Abelian group of an arithmetical structure.

What is an arithmetical structure? Sponsored by the Meyer Fund