Schur Functors and Categorified Plethysm

Abstract

It is known that the Grothendieck group of the category of Schur functors is the ring of symmetric functions. This ring has a rich structure, much of which is encapsulated in the fact that it is a 'plethory': a monoid in the category of birings with its substitution monoidal structure. We show that similarly the category of Schur functors is a '2-plethory', which descends to give the plethory structure on symmetric functions. Thus, much of the structure of symmetric functions exists at a higher level in the category of Schur functors.

Cellular subalgebras of the partition algebra

“…In this section, we discuss just a few generalizations of the partition algebra, although there are indubitably many more than we discuss here. We also give an alternative perspective using tensor categories, which are often strongly linked to combinatorics (which can be seen in, e.g., [BMT21]). 5.1.…”

Indecomposable objects in Khovanov–Sazdanovic's generalizations of Deligne's interpolation categories

“…In characteristic 0, where the idempotents used in the process are Young symmetrizers, the resulting functors are usually called Schur functors (see [FH91,Sec. 6.1], or [BMT21] for a more recent categorical construction of Schur functors).…”

Differential 2-rigs

“…3 Elsewhere, terms like '2-rig' or 'rig categories' have been appropriated by different authors to mean different things. For example, [3] defines a 2-rig to be a cocomplete symmetric monoidal category in which the monoidal product distributes over all colimits, and in [4], '2-rig' has meant a Vect-enriched symmetric monoidal category with biproducts and idempotent splittings (where the distributivity is automatic). On the other hand, the term 'rig category' or 'distributive category' [36] has been used to mean a category with two monoidal products, one called 'multiplicative', which distributes over the other, called 'additive'.…”