Anne Schilling

University of California, Davis

How to get to the moon playing the 15 puzzle

Wednesday, May 15, 2024

6:00–7:00PM

008 Kemeny

Abstract: Probability is the mathematical study of how likely an event occurs or a proposition is true. Representation theory is the study of algebraic structures by realizing their elements as linear maps on vector spaces or modules and decomposing them into their smallest constituents. Both probability and representation theory lend themselves to combinatorial analysis. In this talk we explore how to exploit combinatorial tools (similar to the 15 puzzle) to answer deep questions in probability and representation theory, in particular those that helped to get mankind to the moon.

The ubiquity of crystal bases

Colloquium

Thursday, May 16, 2024

3:15–4:15PM

007 Kemeny

Abstract: Crystal bases are combinatorial skeletons of Lie algebra representations. They appeared in the work of Kashiwara, Lusztig and Littelmann on quantum groups and the geometry of flag varieties. Crystal bases arise in many unexpected places, from mathematical physics to probability and number theory. In this talk, I will showcase ten reasons and applications of how crystal theory can be used to solve problems in representation theory, geometry and beyond.

From quasi-symmetric to Schur expansions via the inverse quasi-Kostka matrix

Seminar talk

Friday, May 17, 2024

3:15–4:15PM

307 Kemeny

Abstract: It is an important problem in algebraic combinatorics to deduce the Schur function expansion of a symmetric function, whose expansion in terms of the fundamental quasisymmetric function is known. For example, formulas are known for the fundamental expansion of a Macdonald symmetric function and for the plethysm of two Schur functions, while the Schur expansions of these expressions are still elusive. Egge, Loehr and Warrington provided a method to obtain the Schur expansion from the fundamental expansion by replacing each quasisymmetric function by a Schur function (not necessarily indexed by a partition) and using straightening rules to obtain the Schur expansion. Here we provide a new method which only involves the coefficients of the quasisymmetric functions indexed by partitions and the quasi-Kostka matrix. As an application, we prove the lexicographically largest term in the Schur expansion of the plethysm of two Schur functions and the Schur expansion of $s_w[s_h](x,y)$ for $w=2,3,4$ using novel symmetric chain decompositions of Young's lattice for partitions in a $w\times h$ box. This is based on joint work with Rosa Orellana, Franco Saliola and Mike Zabrocki.