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Session S16 - Quantum symmetries

Friday, July 16, 11:00 ~ 11:25 UTC-3

Frobenius exact symmetric tensor categories

Pavel Etingof

MIT, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakaca6cbd34b2807458f14ea60a9fda5e1').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addyaca6cbd34b2807458f14ea60a9fda5e1 = '&#101;t&#105;ng&#111;f' + '&#64;'; addyaca6cbd34b2807458f14ea60a9fda5e1 = addyaca6cbd34b2807458f14ea60a9fda5e1 + 'm&#97;th' + '&#46;' + 'm&#105;t' + '&#46;' + '&#101;d&#117;'; var addy_textaca6cbd34b2807458f14ea60a9fda5e1 = '&#101;t&#105;ng&#111;f' + '&#64;' + 'm&#97;th' + '&#46;' + 'm&#105;t' + '&#46;' + '&#101;d&#117;';document.getElementById('cloakaca6cbd34b2807458f14ea60a9fda5e1').innerHTML += '<a ' + path + '\'' + prefix + ':' + addyaca6cbd34b2807458f14ea60a9fda5e1 + '\'>'+addy_textaca6cbd34b2807458f14ea60a9fda5e1+'<\/a>';

I will report on a joint work with K. Coulembier and V. Ostrik. We show that a symmetric tensor category in characteristic p>0 admits a fiber functor to the Verlinde category (semisimplification of Rep$(Z/p)$) if and only if it has moderate growth and its Frobenius functor (an analog of the classical Frobenius in the representation theory of algebraic group) is exact. For example, for p=2 and 3 this implies that any such category is (super)-Tannakian. We also give a characterization of super-Tannakian categories for p>3. This generalizes Deligne's theorem that any symmetric tensor category over C of moderate growth is super-Tannakian to characteristic p. At the end I'll discuss applications of this result to modular representation theory.

Joint work with Kevin Coulembier and Victor Ostrik.