2020 CMS Winter Meeting

Montreal, December 4 - 7, 2020   Algebraic Combinatorixx (Women in Algebraic Combinatorics)
Org: Angele Foley (Laurier) and Steph van Willigenburg (UBC)
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SUNITA CHEPURI, University of Michigan

SAMANTHA DAHLBERG, Arizona State University

PAMELA HARRIS, Williams College
Kostant's partition function and magic multiplex juggling sequences  [PDF]

Kostant’s partition function is a vector partition function that counts the number of ways one can express a weight of a Lie algebra g as a nonnegative integral linear combination of the positive roots of g. Multiplex juggling sequences are generalizations of juggling sequences that specify an initial and terminal configuration of balls and allow for multiple balls at any particular discrete height. Magic multiplex juggling sequences generalize further to include magic balls, which cancel with standard balls when they meet at the same height. In this talk, we present a combinatorial equivalence between positive roots of a Lie algebra and throws during a juggling sequence. This provides a juggling framework to calculate Kostant’s partition functions, and a partition function framework to compute the number of juggling sequences. This is joint work with Carolina Benedetti, Christopher R. H. Hanusa, Alejandro Morales, and Anthony Simpson.

OLYA MANDELSHTAM, Brown University

LUCY MARTINEZ, Stockton University
Minimum Rank of Regular Bipartite Graphs  [PDF]

The rank of a graph $G$ is defined as the rank of its adjacency matrix $A$. The smallest rank among all the matrices with the same pattern of non-zeros entries as $A$, over the field $\mathbb{F}$, is called the minimum rank of $A$ over $\mathbb{F}$. The smallest among all the minimum ranks of $A$ (considering all the fields) is called the minimum rank of $G$. In this work, we study regular bipartite graphs. Specifically, we used linear recursions with linear complexity 2 and zero forcing sets to prove that the minimum rank of a $(n-1)$-regular bipartite graph, with $n$ vertices on each side, is 4.

ROSA ORELLANA, Dartmouth College

ANNA PUN, University of Virginia

SOPHIE SPIRKL, University of Waterloo
A complete multipartite basis for the chromatic symmetric function  [PDF]

The complete multipartite basis $r_\lambda$ for symmetric functions was introduced by Penaguiao. In this talk, I will tell you why this basis is interesting, and give a combinatorial interpretation for the $r_\lambda$-coefficients of the chromatic symmetric function.

Joint work with Logan Crew.

NANCY WALLACE, UQAM
Toward a Schurification of Schröder path formulas.  [PDF]

The Shuffle theorem of Carlsson and Mellit, states that $\nabla(e_n)$ is given by Parking function formulas. Schröder paths are a particular case of Parking functions. These formulas are symmetric in the variables $q$ and $t$. More preciously, for all $n$, $\nabla(e_n)$ can be seen as a $GL_2\times \mathbb{S}_n$-module. In this talk we will put forth a partial formula for the irreducible bicharacters of these modules. Namely we will write subsets of the Schröder paths formulas as products of Schur functions in the variables $q$ and $t$ and the usual Schur functions in the variables $X=\{x_1, x_2, \ldots\}$.