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SUMMARY:Kirillov's orbit method and polynomiality of the faithful dimensio
 n of $p$-groups - Mohammad Bardestani (University of Cambridge)
DTSTART:20180119T140000Z
DTEND:20180119T150000Z
UID:TALK98317@talks.cam.ac.uk
CONTACT:Maurice Chiodo
DESCRIPTION:Let $G$ be a finite group. The faithful dimension of $G$ is de
 fined to be the smallest possible dimension for a faithful complex represe
 ntation of $G$. Aside from its intrinsic interest\, the problem of determi
 ning the faithful dimension of $p$-groups is motivated by its connection t
 o the theory of essential dimension. In this talk\, we will address this p
 roblem for groups of the form $\\mathbf{G}_p:=\\exp(\\mathfrak{g} \\otimes
 _{\\mathbb{Z}}\\mathbb{F}_p)$\, where $\\mathfrak{g}$ is a nilpotent $\\ma
 thbb{Z}$-Lie algebra of finite rank\, and $\\mathbf{G}_p$ is the $p$-group
  associated to $\\mathfrak{g} \\otimes_{\\mathbb{Z}}\\mathbb{F}_p$ in the 
 Lazard correspondence. We will show that in general the faithful dimension
  of $\\mathbf{G}_p$ is given by a finite set of polynomials associated to 
 a partition of the set of prime numbers into Frobenius sets. At the same t
 ime\, we will show that for many naturally arising groups\, including a va
 st class of groups defined by partial orders\, the faithful dimension is g
 iven by a single polynomial. The arguments are reliant on various tools fr
 om number theory\, model theory\, combinatorics and Lie theory.
LOCATION:CMS\, MR13
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