161 research outputs found
On the structure of the centralizer of a braid
The mixed braid groups are the subgroups of Artin braid groups whose elements
preserve a given partition of the base points. We prove that the centralizer of
any braid can be expressed in terms of semidirect and direct products of mixed
braid groups. Then we construct a generating set of the centralizer of any
braid on n strands, which has at most k(k+1)/2 elements if n=2k, and at most
$k(k+3)/2 elements if n=2k+1. These bounds are shown to be sharp, due to work
of N.V.Ivanov and of S.J.Lee. Finally, we describe how one can explicitly
compute this generating set.Comment: Section 5.3 is rewritten. The proposed generating set is shown not to
be minimal, even though it is the smallest one reflecting the geometric
approach. Proper credit is given to the work of other researchers, notably to
N.V.Ivano
Computations for Coxeter arrangements and Solomon's descent algebra II: Groups of rank five and six
In recent papers we have refined a conjecture of Lehrer and Solomon
expressing the character of a finite Coxeter group acting on the th
graded component of its Orlik-Solomon algebra as a sum of characters induced
from linear characters of centralizers of elements of . Our refined
conjecture relates the character above to a component of a decomposition of the
regular character of related to Solomon's descent algebra of . The
refined conjecture has been proved for symmetric and dihedral groups, as well
as finite Coxeter groups of rank three and four.
In this paper, the second in a series of three dealing with groups of rank up
to eight (and in particular, all exceptional Coxeter groups), we prove the
conjecture for finite Coxeter groups of rank five and six, further developing
the algorithmic tools described in the previous article. The techniques
developed and implemented in this paper provide previously unknown
decompositions of the regular and Orlik-Solomon characters of the groups
considered.Comment: Final Version. 17 page
On bounding the bandwidth of graphs with symmetry
We derive a new lower bound for the bandwidth of a graph that is based on a
new lower bound for the minimum cut problem. Our new semidefinite programming
relaxation of the minimum cut problem is obtained by strengthening the known
semidefinite programming relaxation for the quadratic assignment problem (or
for the graph partition problem) by fixing two vertices in the graph; one on
each side of the cut. This fixing results in several smaller subproblems that
need to be solved to obtain the new bound. In order to efficiently solve these
subproblems we exploit symmetry in the data; that is, both symmetry in the
min-cut problem and symmetry in the graphs. To obtain upper bounds for the
bandwidth of graphs with symmetry, we develop a heuristic approach based on the
well-known reverse Cuthill-McKee algorithm, and that improves significantly its
performance on the tested graphs. Our approaches result in the best known lower
and upper bounds for the bandwidth of all graphs under consideration, i.e.,
Hamming graphs, 3-dimensional generalized Hamming graphs, Johnson graphs, and
Kneser graphs, with up to 216 vertices
A New Existence Proof for Ly, the Sporadic Simple Group of R. Lyons
AbstractThis paper reports on a new and independent existence proof for the sporadic simple group Ly of Lyons, using only two permutations of degree 9 606 125, computed by Cooperman, Finkelstein, Tselman, and York. We will show that these two permutations generate a group GâLy, by first computing a base and strong generating set for G, and then checking the two hypotheses for Ly from Lyonsâ original paper. Moreover, this produces a new presentation for Ly
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