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SUMMARY:Lie theory in tensor categories with applications to modular repre
 sentation theory - Pavel Etingof (Massachusetts Institute of Technology)
DTSTART:20240618T124500Z
DTEND:20240618T134500Z
UID:TALK217366@talks.cam.ac.uk
DESCRIPTION:Let G be a group and k an algebraically closed field of charac
 teristic p>0. Let V be a finite dimensional representation of G over k. Th
 en by the classical Krull-Schmidt theorem\, the tensor power V^n can be un
 iquely decomposed into a direct sum of indecomposable representations. But
  we know very little about this decomposition\, even for very small groups
 \, such as G=(Z/2)^3 for p=2 or G=(Z/3)^2 for p=3. For example\, what can 
 we say about the number d_n(V) of such summands of dimension coprime to p?
  It is easy to show that there exists a finite limit d(V) of the n-th root
  of d_n(V)\, but what kind of number is it? For example\, is it algebraic 
 or transcendental? Until recently\, there was no techniques to solve such 
 questions (and in particular the same question about the sum of dimensions
  of these summands is still wide open). Remarkably\, a new subject which m
 ay be called ``Lie theory in tensor categories" gives methods to show that
  d(V) is indeed an algebraic number\, in fact one of a very specific form.
  Moreover\, d is a character of the Green ring of G over k. Finally\, d_n(
 V)>C(V)d(V)^n\, and we can give lower bounds for the constant C(V). In the
  talk I will explain what Lie theory in tensor categories is and how it ca
 n be applied to such problems. This is joint work with K. Coulembier and V
 . Ostrik. &nbsp\;
LOCATION:External
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