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SUMMARY:Pavel Etingof (Massachusetts Institute of Technology)
DTSTART:20230712T160000Z
DTEND:20230712T170000Z
DTSTAMP:20260423T021449Z
UID:TQFT/90
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/TQFT/90/">Li
 e theory in tensor categories (with applications to modular representation
  theory)</a>\nby Pavel Etingof (Massachusetts Institute of Technology) as 
 part of Topological Quantum Field Theory Club (IST\, Lisbon)\n\n\nAbstract
 \nLet $G$ be a group and $k$ an algebraically closed field of characterist
 ic $p$. If $V$ is a finite-dimensional representation of $G$ over $k$\, th
 en by the classical Krull–Schmidt theorem\, the $n$th tensor power of $V
 $ can be uniquely decomposed into a direct sum of indecomposable represent
 ations. But we know very little about this decomposition\, even for very s
 mall groups\, such as $G = (\\Bbb Z/2)^3$ for $p = 2$ or $G = (\\Bbb Z/3)^
 2$ for $p = 3$.\n\nFor example\, what can we say about the number $d_n(V)$
  of summands with dimension coprime to $p$? It is easy to show that there 
 is a finite limit $d(V) := \\lim_{n \\to \\infty} d_n(V)^{1/n}$\, but what
  kind of number is this? Is it algebraic or transcendental? Until recently
 \, there were 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 may be called "Lie theory in tens
 or categories" gives methods to show that $d(V)$ is indeed an algebraic nu
 mber\, which moreover has the form\n\\[ d(V) = \\sum_{1 \\leq j \\leq p/2}
  n_j(V)[j]_q\, \\]\nwhere $n_j(V)$ is a natural number\, $q := \\exp(\\pi 
 i/p)$ is a particular root of unity\, and $[j]_q := \\frac{q^j-q^{-j}}{q-q
 ^{-1}}$ is a $q$-number. Moreover\, $d(V \\oplus W) = d(V) + d(W)$ and $d(
 V \\otimes W) = d(V) d(W)$\, so $d$ is a character of the Green ring of $G
 $ over $k$. Finally\, $d_n(V) \\geq C_V d(V)^n$\, for some $0 < C_V \\leq 
 1$\, and we can give lower bounds for $C_V$. In the talk\, I will explain 
 what Lie theory in tensor categories is and how it can be applied to such 
 problems. This is joint work with K. Coulembier and V. Ostrik.\n
LOCATION:https://researchseminars.org/talk/TQFT/90/
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