Algebraic and geometric methods in combinatorics II
Org: Christophe Hohlweg
(Université du Québec à Montréal)
- MARIA MONKS GILLESPIE, UC Davis
Characterization of queer supercrystals [PDF]
We provide a combinatorial characterization of the crystal bases for the quantum queer superalgebra recently introduced by Grantcharov et al. This characterization is a combination of the local queer relations recently introduced by Assaf and Oguz, with further axioms and a new graph $G$ characterizing the relations between the type $A$ components of the queer crystal. We obtain a combinatorial description of the graph $G$ on the type $A$ components by providing explicit rules for the odd queer operators in terms of simple operations on highest weight words.
This is joint work with Graham Hawkes, Wencin Poh, and Anne Schilling.
- BRENDON RHOADES, UCSD
Spanning configurations [PDF]
Let $V$ be a finite-dimensional vector space. A sequence $(W_1, W_2, \dots, W_r)$ of subspaces of $V$ is a spanning configuration if $V = W_1 + W_2 + \cdots + W_r$ as vector spaces. We will discuss algebraic and combinatorial properties of spanning configurations. Joint with Brendan Pawlowski and Andy Wilson.
- STEPHANIE VAN WILLIGENBURG, UBC
Noncommutative chromatic symmetric functions revisited [PDF]
In 1995 Stanley introduced a generalization of the chromatic polynomial of a graph $G$, called the chromatic symmetric function, $X_G$, which was generalized to noncommuting variables, $Y_G$, by Gebhard-Sagan in 2001. Recently there has been a renaissance in the study of $X_G$, in particular classifying when $X_G$ is a positive linear combination of elementary symmetric or Schur functions, that is, $e$-positive or Schur-positive. In this talk we will extend these new results from $X_G$ to $Y_G$, including classifying when $Y_G$ is noncommutative $e$-positive or noncommutative Schur-positive, in the sense of Bergeron-Hohlweg-Rosas-Zabrocki.
This is joint work with Samantha Dahlberg.
- NATHAN WILLIAMS, UT Dallas
Certain classical generating functions for elements of reflection groups can be expressed using fundamental invariants called exponents. We give new analogues of such generating functions that accommodate orbits of reflecting hyperplanes using similar invariants we call reflexponents. Our verifications are case-by-case.