Massachusetts Institute of Technology, USA
November 26, 2020
Representation theory without vector spaces: A modern view of representation theory is that it is a study not just of individual representations (say, finite dimensional representations of an affine (super)group scheme $G$ over an algebraically closed field $k$) but also of the category Rep$(G)$ formed by them. The properties of Rep$(G)$ can be summarized by saying that it is a symmetric tensor category (shortly, STC) which uniquely determines $G$. It is therefore natural to ask: does the study of STC reduce to group representation theory, or is it more general? In other words, do there exist STC other than Rep$(G)$? If so, this would be interesting, since one can do algebra in any STC, and in categories other than Rep$(G)$ this would be a new kind of algebra.
The answer turns out to be ``yes'', and beautiful examples in characteristic zero were provided by Deligne-Milne in 1981. These very interesting categories are interpolations of representation categories of classical groups GL$(n)$, O$(n)$, Sp$(n)$ to arbitrary values of $n$ in $k$. Deligne later generalized them to symmetric groups and also to characteristic $p$, where, somewhat unexpectedly, one needs to interpolate $n$ to $p$-adic integer values rather than elements of $k$. All these categories violate an obvious necessary condition for a STC to have any realization by finite dimensional vector spaces (and in particular to be of the form $\mathrm{Rep}(G)$): for each object $X$ the length of the $n$-th tensor power of $X$ grows at most exponentially with $n.$ We call this property `moderate growth'. So it is natural to ask if there exist STC of moderate growth other than $\mathrm{Rep}(G)$.
A remarkable theorem of Deligne (2002) says that in characteristic zero, the answer is `no': any such category is of the form $\mathrm{Rep}(G)$, where $G$ is an affine supergroup scheme; in other words, it can be realized in supervector spaces. In particular, algebra in these categories
is just the usual one with equivariance under some supergroup $G$.