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SUMMARY:Georgia Benkart (University of Wisconsin-Madison)
DTSTART:20201003T170000Z
DTEND:20201003T180000Z
DTSTAMP:20260423T024653Z
UID:LieTheory/5
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/LieTheory/5/
 ">Fusion Rules</a>\nby Georgia Benkart (University of Wisconsin-Madison) a
 s part of CRM-Regional Conference in Lie Theory\n\nLecture held in Virtual
 .\n\nAbstract\nFusion rules encode information about tensoring different t
 ypes of modules \n(simple\, projective)  with a finite-dimensional module 
 V\,  and this information\ncan be recorded in a matrix that depends on V. 
  When the objects are complex \nsimple modules for a finite group\, the re
 sulting matrix (often called the McKay \nmatrix due to inspiration from th
 e McKay correspondence)\, has as its right \neigenvectors the columns of t
 he character table of the group\, and the eigenvalues\nare the character o
 f V evaluated on conjugacy class representatives.  So in that\nparticular 
 case\, the eigenvectors are independent of V.   We consider extensions of 
 \nsuch results to other settings  where tensor products are defined such a
 s finite-dimensional\nHopf algebras (e.g. quantum groups at roots of unity
 \, restricted enveloping algebras\nof Lie algebras in prime characteristic
 \, and Drinfeld doubles). The eigenvectors and \neigenvalues have connecti
 ons with the characters of the Hopf algebra\, and in some \nexamples\, man
 y connections with Chebyshev polynomials of various kinds. Fusion rule \nm
 atrices have applications to chip-firing dynamics and to Markov chains.\n\
 nIn this talk I will explain a joint work with Javier Aramayona\, Julio Ar
 oca\, Rachel Skipper and Xiaolei Wu. We define a new family of groups that
  are subgroups of the mapping class group $Map(\\Sigma_g)$ of a surface $\
 \Sigma_g$ of genus $g$ with a Cantor set removed and we will call these gr
 oups Block Mapping Class Groups $B(H)$\, where $H$ is a subgroup of $\\Sig
 ma_g$ . More visually\, this family will be constructed by making a tree-l
 ike surface gluing pair of pants and taking homeomorphisms that depend on 
 $H$ with certain preservation properties (it will preserve what we will ca
 ll a block decomposition of this surface\, hence the name of our groups). 
 We will see that this family is closely related to Thompson’s groups and
  that it has the property of being of type $F_n$ if and only if $H$ is. As
  a consequence\, for every $g\\in \\mathbb N \\cup \\{0\, \\infty\\}$ and 
 every $n\\ge 1$\, we construct a subgroup $G <\\Map(\\Sigma_g)$ that is of
  type $F_n$ but not of type $F_{n+1}$\, and which contains the mapping cla
 ss group of every compact surface of genus less or equal to $\\g$ and with
  non-empty boundary. As expected in this workshop\, the techniques involve
  manipulating cube complexes\, as the Stein-Farley cube complex.\n
LOCATION:https://researchseminars.org/talk/LieTheory/5/
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