Combinatorics & Algebra Seminar-Categorifying the plethysm product on symmetric functions

Title: Categorifying the plethysm product on symmetric functions

Speaker: Joe Moeller, NIST
Date and time: Friday, February 18, 3–4 pm
Place: Phillips 736

Abstract: It is well-known that representations of symmetric groups are classified by symmetric functions. The ring of symmetric functions enjoys a rich algebraic and coalgebraic structure, in particular a co-ring structure as well as an extra operation known as the substitution or plethysm product. This structure has been studied by Atiyah--Tall, Tall--Wraith, and Borger--Wieland in an effort to further understand λ-rings and Adams operations in K-theory. We seek to shed light on this structure by giving a categorical account of its origin. We demonstrate that an analogous structure arises in a nearly trivial way (from a certain perspective) on the category of Schur functors. We then show that this determines the co-ring and plethysm structure on the ring of symmetric functions for purely categorical reasons. In this way, properties of these structures are derived without direct reference to their formidable formulas.