Monoid representations and partitions (Charlotte Aten at DU Algebra and Logic Seminar 2023)
Channel: Charlotte Aten
Date: 2024-02-10

This is from my old YouTube channel. My new channel can be found at https://www.youtube.com/@CharlotteAten

This is a talk that I gave for the University of Denver Algebra and Logic Seminar in the fall of 2023. I discuss a formula I found when I was a graduate student which involves the partition numbers. During the summer of 2023 I was reminded of this and posted a preprint to the arXiv (linked below) which I mostly wrote back then. The formula is obtained by using Burnside's Lemma in the context of monoid representations on a finite set.

Title: Monoid representations and partitions
Abstract: When I began graduate school, I was interested in the representation theory of monoids, so I studied the set representations of the free idempotent monoid on one generator. These are nothing more than the idempotent maps from the set to itself. The collection of such maps carries a natural group action. By applying Burnside's Lemma this leads to a formula which only involves a partition number and elementary functions. One side involves a summation over a set closely related to the partition number, however. Some speculation is made as to how to eliminate this summation.

Slides: https://aten.cool/documents/aten_du_algebra_logic_2023.pdf
Paper: https://arxiv.org/abs/2308.10177