742 research outputs found
Homology representations arising from the half cube, II
In a previous work (arXiv:0806.1503v2), we defined a family of subcomplexes
of the -dimensional half cube by removing the interiors of all half cube
shaped faces of dimension at least , and we proved that the homology of such
a subcomplex is concentrated in degree . This homology group supports a
natural action of the Coxeter group of type . In this paper, we
explicitly determine the characters (over ) of these homology
representations, which turn out to be multiplicity free. Regarded as
representations of the symmetric group by restriction, the homology
representations turn out to be direct sums of certain representations induced
from parabolic subgroups. The latter representations of \sym_n agree (over
) with the representations of \sym_n on the -nd homology of
the complement of the -equal real hyperplane arrangement.Comment: 19 pages AMSTeX. One figure. The Conjecture in the previous version
is now a Theorem. This research was supported by NSF grant DMS-090576
Geometrical Frustration and Static Correlations in Hard-Sphere Glass Formers
We analytically and numerically characterize the structure of hard-sphere
fluids in order to review various geometrical frustration scenarios of the
glass transition. We find generalized polytetrahedral order to be correlated
with increasing fluid packing fraction, but to become increasingly irrelevant
with increasing dimension. We also find the growth in structural correlations
to be modest in the dynamical regime accessible to computer simulations.Comment: 21 pages; part of the "Special Topic Issue on the Glass Transition
K-area, Hofer metric and geometry of conjugacy classes in Lie groups
Given a closed symplectic manifold we introduce a certain
quantity associated to a tuple of conjugacy classes in the universal cover of
the group by means of the Hofer metric on
. We use pseudo-holomorphic curves involved in the
definition of the multiplicative structure on the Floer cohomology of a
symplectic manifold to estimate this quantity in terms of actions
of some periodic orbits of related Hamiltonian flows. As a corollary we get a
new way to obtain Agnihotri-Belkale-Woodward inequalities for eigenvalues of
products of unitary matrices. As another corollary we get a new proof of the
geodesic property (with respect to the Hofer metric) of Hamiltonian flows
generated by certain autonomous Hamiltonians. Our main technical tool is K-area
defined for Hamiltonian fibrations over a surface with boundary in the spirit
of L.Polterovich's work on Hamiltonian fibrations over .Comment: Corrected final version, accepted for publication in Inventiones
Mathematica
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