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SUMMARY:Pavel Etingof (MIT)
DTSTART:20201117T160000Z
DTEND:20201117T165000Z
DTSTAMP:20260419T112357Z
UID:icra2020/11
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/icra2020/11/
 ">Symmetric tensor categories III</a>\nby Pavel Etingof (MIT) as part of I
 CRA 2020\n\n\nAbstract\nLecture 3. Symmetric tensor categories of moderate
  growth and modular representation theory.\n\nDeligne categories discussed
  in Lecture 2 violate an obvious necessary condition for a symmetric tenso
 r category (STC) to have any realization by finite dimensional vector spac
 es (and in particular to be of the form Rep(G) ): for each object X the le
 ngth of the n -th tensor power of X grows at most exponentially with n . W
 e call this property “moderate growth”. So it is natural to ask if the
 re exist STC of moderate growth other than Rep(G) . In characteristic zero
 \, the negative answer is given by the remarkable theorem of Deligne (2002
 )\, discussed in Lecture 1. Namely Deligne’s theorem says that a STC of 
 moderate growth can always be realized in supervector spaces. However\, in
  characteristic p the situation is much more interesting. Namely\, Deligne
 ’s theorem is known to fail in any characteristic p>0 . The simplest exo
 tic symmetric tensor category of moderate growth (i.e.\, not of the form R
 ep(G) ) for p>3 is the semisimplification of the category of representatio
 ns of Z/p \, called the Verlinde category. For example\, for p=5 \, this c
 ategory has an object X such that X2=X+1 \, so X cannot be realized by a v
 ector space (as its dimension would have to equal the golden ratio). I wil
 l discuss some aspects of algebra in these categories\, in particular fail
 ure of the PBW theorem for Lie algebras (and how to fix it) and Ostrik’s
  generalization of Deligne’s theorem in characteristic p . I will also d
 iscuss a family of non-semisimple exotic categories in characteristic p co
 nstructed in my joint work with Dave Benson and Victor Ostrik\, and their 
 relation to the representation theory of groups (Z/p)n over a field of cha
 racteristic p .\n
LOCATION:https://researchseminars.org/talk/icra2020/11/
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