520 research outputs found
Bounds for complexity of syndrome decoding for poset metrics
In this work we show how to decompose a linear code relatively to any given
poset metric. We prove that the complexity of syndrome decoding is determined
by a maximal (primary) such decomposition and then show that a refinement of a
partial order leads to a refinement of the primary decomposition. Using this
and considering already known results about hierarchical posets, we can
establish upper and lower bounds for the complexity of syndrome decoding
relatively to a poset metric.Comment: Submitted to ITW 201
Coxeter transformation groups and reflection arrangements in smooth manifolds
Artin groups are a natural generalization of braid groups and are
well-understood in certain cases. Artin groups are closely related to Coxeter
groups. There is a faithful representation of a Coxeter group as a linear
reflection group on a real vector space . The group acts properly and fixes
a union of hyperplanes. The -action extends as the covering space action to
the complexified complement of these hyperplanes. The fundamental groups of the
complement and the orbit space are the pure Artin group and the Artin group
respectively. For the Artin groups of finite type Deligne proved that the
associated complement is aspherical. Using the Coxeter group data Salvetti gave
a construction of a cell complex which is a -equivariant deformation retract
of the complement. This construction was independently generalized by Charney
and Davis to the Artin groups of infinite type. A lot of algebraic properties
of these groups were discovered using combinatorial and topological properties
of this cell complex.
In this paper we represent a Coxeter group as a subgroup of diffeomorphisms
of a smooth manifold. These so-called Coxeter transformation groups fix a union
of codimension- (reflecting) submanifolds and permute the connected
components of the complement. Their action naturally extends to the tangent
bundle of the ambient manifold and fixes the union of tangent bundles of these
reflecting submanifolds. Fundamental group of the tangent bundle complement and
that of its quotient serve as the analogue of pure Artin group and Artin group
respectively. The main aim of this paper is to prove Salvetti's theorems in
this context. We show that the combinatorial data of the Coxeter transformation
group can be used to construct a cell complex homotopy equivalent to the
tangent bundle complement and that this homotopy equivalence is equivariant.Comment: 18 pages, 2 figures. V2: minor changes, typos fixed. final versio
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