7,145 research outputs found
Covering With Tensor Products and Powers
We study when a tensor product of irreducible representations of the
symmetric group contains all irreducibles as subrepresentations - we say
such a tensor product covers . Our results show that this behavior
is typical. We first give a general criterion for such a tensor product to have
this property. Using this criterion we show that the tensor product of a
constant number of random irreducibles covers asymptotically
almost surely. We also consider, for a fixed irreducible representation, the
degree of tensor power needed to cover . We show that the simple
lower bound based on dimension is tight up to a universal constant factor for
every irreducible representation, as was recently conjectured by Liebeck,
Shalev, and Tiep
Gaussian fluctuations of Young diagrams and structure constants of Jack characters
In this paper, we consider a deformation of Plancherel measure linked to Jack
polynomials. Our main result is the description of the first and second-order
asymptotics of the bulk of a random Young diagram under this distribution,
which extends celebrated results of Vershik-Kerov and Logan-Shepp (for the
first order asymptotics) and Kerov (for the second order asymptotics). This
gives more evidence of the connection with Gaussian -ensemble, already
suggested by some work of Matsumoto.
Our main tool is a polynomiality result for the structure constant of some
quantities that we call Jack characters, recently introduced by Lassalle. We
believe that this result is also interested in itself and we give several other
applications of it.Comment: 71 pages. Minor modifications from version 1. An extended abstract of
this work, with significantly fewer results and a different title, is
available as arXiv:1201.180
A robust method for cluster analysis
Let there be given a contaminated list of n R^d-valued observations coming
from g different, normally distributed populations with a common covariance
matrix. We compute the ML-estimator with respect to a certain statistical model
with n-r outliers for the parameters of the g populations; it detects outliers
and simultaneously partitions their complement into g clusters. It turns out
that the estimator unites both the minimum-covariance-determinant rejection
method and the well-known pooled determinant criterion of cluster analysis. We
also propose an efficient algorithm for approximating this estimator and study
its breakdown points for mean values and pooled SSP matrix.Comment: Published at http://dx.doi.org/10.1214/009053604000000940 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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