Schur Functors and Categorified Plethysm

2021/06/01 by John C. Baez, Baez, John C., Joe Moeller +3 · 2 voices
#math.RT #math.CT

paper · pdf · doi:10.48550/arxiv.2106.00190

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.

Discussions

• Thanks! Here it is, in case anyone cares. The ring of symmetric functions has lots of structure including a plethysm product. But it's just the Grothendieck group of an interesting category, and we sh [bsky, 6 points, 2 comments]
• plethory structure on its Grothendieck group, which happens to be the ring of symmetric functions. I think this is somehow very close to what's happening in Tannaka reconstruction, but I don't know it [bsky, 1 points, 1 comments]

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