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CATEGORIES:Geometric Group Theory (GGT) Seminar
SUMMARY:Kirillov's orbit method and polynomiality of the f
aithful dimension of $p$-groups - Mohammad Bardest
ani (University of Cambridge)
DTSTART;TZID=Europe/London:20180119T140000
DTEND;TZID=Europe/London:20180119T150000
UID:TALK98317AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/98317
DESCRIPTION:Let $G$ be a finite group. The faithful dimension
of $G$ is defined to be the smallest possible dime
nsion for a faithful complex representation of $G$
. Aside from its intrinsic interest\, the problem
of determining the faithful dimension of $p$-group
s is motivated by its connection to the theory of
essential dimension. In this talk\, we will addres
s this problem for groups of the form $\\mathbf{G}
_p:=\\exp(\\mathfrak{g} \\otimes_{\\mathbb{Z}}\\ma
thbb{F}_p)$\, where $\\mathfrak{g}$ is a nilpotent
$\\mathbb{Z}$-Lie algebra of finite rank\, and $\
\mathbf{G}_p$ is the $p$-group associated to $\\ma
thfrak{g} \\otimes_{\\mathbb{Z}}\\mathbb{F}_p$ in
the Lazard correspondence. We will show that in ge
neral the faithful dimension of $\\mathbf{G}_p$ is
given by a finite set of polynomials associated t
o a partition of the set of prime numbers into Fro
benius sets. At the same time\, we will show that
for many naturally arising groups\, including a va
st class of groups defined by partial orders\, the
faithful dimension is given by a single polynomia
l. The arguments are reliant on various tools from
number theory\, model theory\, combinatorics and
Lie theory.
LOCATION:CMS\, MR13
CONTACT:Maurice Chiodo
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