John Baez and James Dolan, 2023-08-17

Recall from last time that N! is another name for the symmetric group S_N, a '2-rig' over a field k is a symmetric monoidal cocomplete k-linear category, and an 'N!-torsor' in a 2-rig R over k is a map of 2-rigs from the 2-rig of representations of N! over k to R.
We are trying to get special commutative Frobenius algebras in a 2-rig R from N!-torsors in R, for example when k = ℚ. Can we describe this process as a 2-rig map from 'the theory of special commutative Frobenius algebras' to 'the theory of N!-torsors'? The theory of N!-torsors is simply the 2-rig of representations of N!. So, we need to be better understand the theory of special commutative Frobenius algebras.
Perhaps we should switch from using 'theories' (i.e. 2-rigs) to props. The prop of special commutative Frobenius algebras is described in Proposition 7.1 here:
Brandon Coya and Brendan Fong, Corelations are the prop for extraspecial commutative Frobenius monoids, arxiv.org/abs/...
To describe N!-torsors we are led to consider the 'Hecke prop', and thus the role of Hecke operators and more general 'Hecke tensors' in this subject.
For background on special commutative Frobenius algebras and separable commutative algebras, see:
Aurelio Carboni, Matrices, relations, and group representations, www.sciencedir...
John Baez, Grothendieck-Galois-Brauer Theory (Part 1), golem.ph.utexa...
For more on this whole series of conversations, go here:
math.ucr.edu/h...