1,706 research outputs found
Alternating subgroups of Coxeter groups
We study combinatorial properties of the alternating subgroup of a Coxeter
group, using a presentation of it due to Bourbaki.Comment: 39 pages, 3 figure
The alternating Hecke algebra and its representations.
The alternating Hecke algebra is a q-analogue of the alternating subgroups of the finite Coxeter groups. Mitsuhashi has looked at the representation theory in the cases of the Coxeter groups of type A_n, and B_n, and here we provide a general approach that can be applied to any finite Coxeter group. We give various bases and a generating set for the alternating Hecke algebra. We then use Tits' deformation theorem to prove that, over a large enough field, the alternating Hecke algebra is isomorphic to the group algebra of the corresponding alternating Coxeter group. In particular, there is a bijection between the irreducible representations of the alternating Hecke algebra and the irreducible representations of the alternating subgroup. In chapter 5 we discuss the branching rules from the Iwahori-Hecke algebra to the alternating Hecke algebra and give criteria that determine these for the Iwahori-Hecke algebras of types A_n, B_n and D_n. We then look specifically at the alternating Hecke algebra associated to the symmetric group and calculate the values of the irreducible characters on a set of minimal length conjugacy class representatives
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Separability within alternating groups and randomness
This thesis promotes known residual properties of free groups, surface groups, right angled Coxeter groups and right angled Artin groups to the situation where the quotient is only allowed to be an alternating group. The proofs follow two related threads of ideas.
The first thread leads to `alternating' analogues of extended residual finiteness in surface groups \cite{scott1978subgroups}, right angled Artin groups and right angled Coxeter groups \cite{haglund2008finite}.
Let be a right-angled Coxeter group corresponding to a finite non-discrete graph with at least vertices. Our main theorem says that is connected if and only if for any infinite index convex-cocompact subgroup of and any finite subset there is a surjective homomorphism from to a finite alternating group such that . A corollary is that a right-angled Artin group splits as a direct product of cyclic groups and groups with many alternating quotients in the above sense.
Similarly, finitely generated subgroups of closed, orientable, hyperbolic surface groups can be separated from finitely many elements in an alternating quotient, answering positively a conjecture of Wilton \cite{wilton2012alternating}.
The second thread uses probabilistic methods to provide `alternating' analogues of subgroup conjugacy separability and subgroup into-conjugacy separability in free groups \cite{bogopolski2010subgroup}.
Suppose are infinite index, finitely generated subgroups of a non-abelian free group . Then there exists a surjective homomorphism such that if is not conjugate into , then is not conjugate into .EPSRC
International Doctoral Scholar schem
The mod 2 cohomology rings of the alternating subgroups of the Coxeter groups of Type B
We show that the direct sum of the cohomology groups of the alternating
subgroups of the family of Coxeter groups of Type B exhibits an almost-Hopf
ring structure. We apply techniques developed by Giusti and Sinha to fully
compute a presentation of this structure for mod 2 coefficient.Comment: 34 pages, 1 figur
The FAn Conjecture for Coxeter groups
We study global fixed points for actions of Coxeter groups on nonpositively
curved singular spaces. In particular, we consider property FA_n, an analogue
of Serre's property FA for actions on CAT(0) complexes. Property FA_n has
implications for irreducible representations and complex of groups
decompositions. In this paper, we give a specific condition on Coxeter
presentations that implies FA_n and show that this condition is in fact
equivalent to FA_n for n=1 and 2. As part of the proof, we compute the
Gersten-Stallings angles between special subgroups of Coxeter groups.Comment: This is the version published by Algebraic & Geometric Topology on 19
November 200
Quasi Regular Polyhedra and Their Duals with Coxeter Symmetries Represented by Quaternions I
In two series of papers we construct quasi regular polyhedra and their duals
which are similar to the Catalan solids. The group elements as well as the
vertices of the polyhedra are represented in terms of quaternions. In the
present paper we discuss the quasi regular polygons (isogonal and isotoxal
polygons) using 2D Coxeter diagrams. In particular, we discuss the isogonal
hexagons, octagons and decagons derived from 2D Coxeter diagrams and obtain
aperiodic tilings of the plane with the isogonal polygons along with the
regular polygons. We point out that one type of aperiodic tiling of the plane
with regular and isogonal hexagons may represent a state of graphene where one
carbon atom is bound to three neighboring carbons with two single bonds and one
double bond. We also show how the plane can be tiled with two tiles; one of
them is the isotoxal polygon, dual of the isogonal polygon. A general method is
employed for the constructions of the quasi regular prisms and their duals in
3D dimensions with the use of 3D Coxeter diagrams.Comment: 22 pages, 16 figure
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