Lectures on symmetric tensor categories (Q6131068)

This is an extended version of the notes by the second author of the lectures on symmetric tensor categories given by the first author at Ohio State University in March 2019 and later at ICRA-2020 in November 2020. The synopsis of the paper goes as follows. \begin{itemize} \item[\S 2] is Lecture 1, reviewing the general theory of symmetric tensor categories and discussing such examples in characteristic zero (so-called Deligne categories) as well as Deligne's theorem [\textit{P. Deligne}, Mosc. Math. J. 2, No. 2, 227--248 (2002; Zbl 1005.18009)]. \item[\S 3] is Lecture 2, reviewing the structure of Deligne categories \(\mathrm{Rep\,}GL_{t}\) in interpolation of the classical representation categories \(\mathrm{Rep\,}GL_{n}\left( \mathbb{C}\right) \) and discussing their alternative construction by use of ultrafilters. \item[\S 4] is Lecture 3, discussing the notion of semisimplification, the Verlinde category, Ostrik's generalization of Deligne's theorem for fusion categories in characteristic \(p\) and recent further extension to semisimple (and, more generally, Frobenius exact) symmetric tensor categories [\textit{K. Coulembier} et al., Ann. Math. (2) 197, No. 3, 1235--1279 (2023; Zbl 07668530)]. The authors also discuss the Verlinde categories for prime powers, \(\mathrm{Ver}_{p^{n}}\), constructed in [\textit{D. Benson} and \textit{P. Etingof}, Adv. Math. 351, 967--999 (2019; Zbl 1430.18013); \textit{D. Benson} et al., Duke Math. J. 172, No. 1, 105--200 (2023; Zbl 1511.18019)] as well as a conjectural generalization of Deligne's theorem to general symmetric tensor categories of moderate growth in characteristic \(p\) which involves \(\mathrm{Ver}_{p^{n}}\). \item[\S 5] is the appendix discussing some applications of these techniques to modular representation theory and studying dimensions in ribbon categories in positive characteristic. \end{itemize} For the entire collection see [Zbl 1530.16002].