Combinatorics/Algebra Seminar
Spring 2019 Archive:

April 29

O'Neill Kingston: Jeux de taquin.

Abstract:

A Young tableau is a combinatorial object comprised of an array of boxes filled with letters from some alphabet. We may define the operations of "bumping" and "sliding" on a tableau, giving rise to a process known as the jeu de taquin ("15 puzzle") with applications in representation theory. I will give an overview of these operations in types A and C.

April 22

Stefanie Wang (Trinity College and Smith College): Counting words in free quasigroups.

Abstract:

A quasigroup is an algebra equipped with three nonassociative binary operations. This talk will discuss the method used to count the number of reduced words in free quasigroups on s generators of a given length n, the so-called peri-Catalan numbers. I will present a recursive formula for the peri-Catalan numbers, based on the Euclidean Algorithm.

April 8, 15

Alex Nowak: Conjugates of quantum quasigroups.

Abstract:

Associated to any quasigroup are six potentially distinct conjugate quasigroups. Coincidence of conjugates is determined by triality (S₃-symmetry). In the theory of quantum quasigroups, an extension of quasigroups to symmetric monoidal categories, the mere existence of conjugates is a nontrivial matter. We will review two classes of quantum quasigroups whose systems of conjugates resemble the classical case:

1. linear quantum quasigroups in categories of modules under direct sum, and
2. cocommutative Hopf algebras.

March 25, April 1

Jonathan Smith: Augmented quasigroups.

Abstract:

Augmented quasigroups provide a unified approach to the study of many different objects that have appeared in the literature, including hypergroups, weighted quasigroups, quasigroups, and Heyting algebras. They are defined in the context of a compact closed category, which encodes data from the duality of finite dimensional vector spaces. In the first talk, we will provide an introduction to these categories, including the important category of relations between sets. Then, in the second talk, we will describe the augmented quasigroups.

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