568 research outputs found
Asymptotical behaviour of roots of infinite Coxeter groups
Let W be an infinite Coxeter group. We initiate the study of the set E of
limit points of "normalized" positive roots (representing the directions of the
roots) of W. We show that E is contained in the isotropic cone of the bilinear
form B associated to a geometric representation, and illustrate this property
with numerous examples and pictures in rank 3 and 4. We also define a natural
geometric action of W on E, and then we exhibit a countable subset of E, formed
by limit points for the dihedral reflection subgroups of W. We explain that
this subset is built from the intersection with Q of the lines passing through
two positive roots, and finally we establish that it is dense in E.Comment: 19 pages, 11 figures. Version 2: 29 pages, 11 figures. Reorganisation
of the paper, addition of many details (section 5 in particular). Version 3 :
revised edition accepted in Journal of the CMS. The number "I" was removed
from the title since number "II" paper was named differently, see
http://arxiv.org/abs/1303.671
Unitary reflection groups for quantum fault tolerance
This paper explores the representation of quantum computing in terms of
unitary reflections (unitary transformations that leave invariant a hyperplane
of a vector space). The symmetries of qubit systems are found to be supported
by Euclidean real reflections (i.e., Coxeter groups) or by specific imprimitive
reflection groups, introduced (but not named) in a recent paper [Planat M and
Jorrand Ph 2008, {\it J Phys A: Math Theor} {\bf 41}, 182001]. The
automorphisms of multiple qubit systems are found to relate to some Clifford
operations once the corresponding group of reflections is identified. For a
short list, one may point out the Coxeter systems of type and (for
single qubits), and (for two qubits), and (for three
qubits), the complex reflection groups and groups No 9 and 31 in
the Shephard-Todd list. The relevant fault tolerant subsets of the Clifford
groups (the Bell groups) are generated by the Hadamard gate, the phase
gate and an entangling (braid) gate [Kauffman L H and Lomonaco S J 2004 {\it
New J. of Phys.} {\bf 6}, 134]. Links to the topological view of quantum
computing, the lattice approach and the geometry of smooth cubic surfaces are
discussed.Comment: new version for the Journal of Computational and Theoretical
Nanoscience, focused on "Technology Trends and Theory of Nanoscale Devices
for Quantum Applications
Semidirect product decomposition of Coxeter groups
Let be a Coxeter system, let be a partition of
such that no element of is conjugate to an element of , let
be the set of -conjugates of elements of and let
be the subgroup of generated by . We show
that and that is
a Coxeter system.Comment: 28 pages, one table. We have added some comments on parabolic
subgroups, double cosets representatives, finite and affine Weyl groups,
invariant theory, Solomon descent algebr
Coxeter Complexes and Graph-Associahedra
Given a graph G, we construct a simple, convex polytope whose face poset is
based on the connected subgraphs of G. This provides a natural generalization
of the Stasheff associahedron and the Bott-Taubes cyclohedron. Moreover, we
show that for any simplicial Coxeter system, the minimal blow-ups of its
associated Coxeter complex has a tiling by graph-associahedra. The geometric
and combinatorial properties of the complex as well as of the polyhedra are
given. These spaces are natural generalizations of the Deligne-Knudsen-Mumford
compactification of the real moduli space of curves.Comment: 18 pages, 9 figures; revised content and reference
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