246 research outputs found
Irreducible Coxeter groups
We prove that a non-spherical irreducible Coxeter group is (directly)
indecomposable and that a non-spherical and non-affine Coxeter group is
strongly indecomposable in the sense that all its finite index subgroups are
(directly) indecomposable. We prove that a Coxeter group has a decomposition as
a direct product of indecomposable groups, and that such a decomposition is
unique up to a central automorphism and a permutation of the factors. We prove
that a Coxeter group has a virtual decomposition as a direct product of
strongly indecomposable groups, and that such a decomposition is unique up to
commensurability and a permutation of the factors
Minimal Faithful Permutation Degrees for Irreducible Coxeter Groups and Binary Polyhedral Groups
In this article we calculate the minimal faithful permutation degree for all
of the irreducible Coxeter groups. We also exhibit new examples of finite
groups that possess a quotient whose minimal degree is strictly greater than
that of the group.Comment: 22 page
On the direct indecomposability of infinite irreducible Coxeter groups and the Isomorphism Problem of Coxeter groups
In this paper we prove, without the finite rank assumption, that any
irreducible Coxeter group of infinite order is directly indecomposable as an
abstract group. The key ingredient of the proof is that we can determine, for
an irreducible Coxeter group, the centralizers of the normal subgroups that are
generated by involutions. As a consequence, we show that the problem of
deciding whether two general Coxeter groups are isomorphic, as abstract groups,
is reduced to the case of irreducible Coxeter groups, without assuming the
finiteness of the number of the irreducible components or their ranks. We also
give a description of the automorphism group of a general Coxeter group in
terms of those of its irreducible components.Comment: 30 page
Polynomial solutions to the WDVV equations in four dimensions
All polynomial solutions of the WDVV equations for the case n = 4 are determined. We find all five solutions predicted by Dubrovin, namely those corresponding to Frobenius structures on orbit spaces of finite Coxeter groups. Moreover we find two additional series of polynomial solutions of which one series is of semi-simple type (massive). This result supports Dubrovin's conjecture if modified appropriately
Involutory reflection groups and their models
AbstractA finite subgroup G of GL(n,C) is involutory if the sum of the dimensions of its irreducible complex representations is given by the number of absolute involutions in the group, i.e. elements gâG such that ggÂŻ=1, where the bar denotes complex conjugation. A uniform combinatorial model is constructed for all non-exceptional irreducible complex reflection groups which are involutory including, in particular, all infinite families of finite irreducible Coxeter groups
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