79 research outputs found
Fusion Systems for Profinite Groups
We introduce the notion of a pro-fusion system on a pro-p group, which
generalizes the notion of a fusion system on a finite p-group. We also prove a
version of Alperin's Fusion Theorem for pro-fusion systems.Comment: 23 page
Connected components of compact matrix quantum groups and finiteness conditions
We introduce the notion of identity component of a compact quantum group and
that of total disconnectedness. As a drawback of the generalized Burnside
problem, we note that totally disconnected compact matrix quantum groups may
fail to be profinite. We consider the problem of approximating the identity
component as well as the maximal normal (in the sense of Wang) connected
subgroup by introducing canonical, but possibly transfinite, sequences of
subgroups. These sequences have a trivial behaviour in the classical case. We
give examples, arising as free products, where the identity component is not
normal and the associated sequence has length 1.
We give necessary and sufficient conditions for normality of the identity
component and finiteness or profiniteness of the quantum component group. Among
them, we introduce an ascending chain condition on the representation ring,
called Lie property, which characterizes Lie groups in the commutative case and
reduces to group Noetherianity of the dual in the cocommutative case. It is
weaker than ring Noetherianity but ensures existence of a generating
representation. The Lie property and ring Noetherianity are inherited by
quotient quantum groups. We show that A_u(F) is not of Lie type. We discuss an
example arising from the compact real form of U_q(sl_2) for q<0.Comment: 43 pages. Changes in the introduction. The relation between our and
Wang's notions of central subgroup has been clarifie
On the cohomology of pro-fusion systems
We prove the Cartan-Eilenberg stable elements theorem and construct a
Lyndon-Hochschild-Serre type spectral sequence for pro-fusion systems. As an
application, we determine the continuous mod- cohomology ring of
for any odd prime .Comment: 19 pages, 4 figure
Finiteness properties of profinite groups
Broadly speaking, a finiteness property of groups is any generalisation of
the property of having finite order. A large part of infinite group theory is
concerned with finiteness properties and the relationships between them.
Profinite groups are an important case of this, being compact topological
groups that possess an intimate connection with their finite images. This
thesis investigates the relationship between several finiteness properties that
a profinite group may have, with consequences for the structure of finite and
profinite groups.Comment: PhD thesis, 127 page
Generalized Zeta function representation of groups and 2-dimensional Topological Yang-Mills theory: The example of GL(2, F_q) and PGL(2, F_q)
We recall the relation between Zeta function representation of groups and
two-dimensional topological Yang-Mills theory through Mednikh formula. We prove
various generalisations of Mednikh formulas and define generalization of Zeta
functions representations of groups. We compute some of these functions in the
case of the finite group and We
recall the table characters of these groups for any , compute the
Frobenius-Schur indicator of their irreducible representations and give the
explicit structure of their fusion ringsComment: 27 page
Wreath products and projective system of non Schurian association schemes
A wreath product is a method to construct an association scheme from two
association schemes. We determine the automorphism group of a wreath product.
We show a known result that a wreath product is Schurian if and only if both
components are Schurian, which yields large families of non-Schurian
association schemes and non-Schurian -rings. We also study iterated wreath
products. Kernel schemes by Martin and Stinson are shown to be iterated wreath
products of class-one association schemes. The iterated wreath products give
examples of projective systems of non-Schurian association schemes, with an
explicit description of primitive idempotents
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