Ohio State University, November 5, 2020

University of Denver, February 4, 2022

John Baez

Schur Functors

The representation theory of the symmetric groups is clarified by thinking of all representations of all these groups as objects of a single category: the category of Schur functors. These play a universal role in representation theory, since Schur functors act on the category of representations of any group. We can understand this as an example of categorification. A "rig" is a "ring without negatives", and the free rig on one generator is \(\mathbb{N}[x]\), the rig of polynomials with natural number coefficients. Categorifying the concept of commutative rig we obtain the concept of "symmetric 2-rig", and it turns out the category of Schur functors is the free symmetric 2-rig on one generator. Thus, in a certain sense, Schur functors are the next step after polynomials.

You can see the slides here, and also a video from when I gave the talk at Ohio State:

For details read our paper:

Also here is a pre-talk, to give people a bit of background for the actual talk:

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