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Title:
Symmetric tensor categories in positive characteristic
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Abstract:
A celebrated theorem of Deligne (2002) says that every symmetric tensor category of moderate growth over $\mathbb C$ is super-Tannakian (i.e., admits a fiber functor to the category of supervector spaces, which means that it is the representation category of a supergroup). But in characteristic $p$ the situation is a lot more complicated and interesting. Namely, the (braided) Verlinde category of integrable representations of the affine Lie algebra $\widehat{\mathfrak sl}_2$ at level $p^n-2$ which occurs in the Wess-Zumino-Witten conformal field theory has a reduction modulo $p$ such that the braiding becomes symmetric. The resulting category ${\rm Ver}_{p^n}$ is semisimple for $n=1$ and was defined in this case by Gelfand-Kazhdan and Georgiev-Mathieu in early 1990s, but for $n>1$ it is not semisimple and was constructed only 2 years ago. The categories ${\rm Ver}_{p^n}$ form a nested sequence and are incompressible (do not admit fiber functors into smaller categories). We conjecture that any symmetric tensor category of moderate growth in characteristic $p$ has a fiber functor into ${\rm Ver}_{p^\infty}=\cup_n {\rm Ver}_{p^n}$. We also show that such a category has a fiber functor to ${\rm Ver}_p$ if and only if it is Frobenius exact, i.e. its Frobenius functor is exact. In particular, this includes semisimple categories, which has powerful applications to modular representation theory of finite groups. Also this reduces the study of Frobenius exact categories of moderate growth to Lie theory in ${\rm Ver}_p$, an exciting new subject of future research already having a number of very interesting results but even more open problems. The talk is based on joint work with D. Benson, K. Coulembier and V. Ostrik,
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