19,286 research outputs found
Effective Invariant Theory of Permutation Groups using Representation Theory
Using the theory of representations of the symmetric group, we propose an
algorithm to compute the invariant ring of a permutation group. Our approach
have the goal to reduce the amount of linear algebra computations and exploit a
thinner combinatorial description of the invariant ring.Comment: Draft version, the corrected full version is available at
http://www.springer.com
Equivariant Ehrhart theory
Motivated by representation theory and geometry, we introduce and develop an
equivariant generalization of Ehrhart theory, the study of lattice points in
dilations of lattice polytopes. We prove representation-theoretic analogues of
numerous classical results, and give applications to the Ehrhart theory of
rational polytopes and centrally symmetric polytopes. We also recover a
character formula of Procesi, Dolgachev, Lunts and Stembridge for the action of
a Weyl group on the cohomology of a toric variety associated to a root system.Comment: 40 pages. Final version. To appear in Adv. Mat
The non-abelian Born-Infeld action at order F^6
To gain insight into the non-abelian Born-Infeld (NBI) action, we study
coinciding D-branes wrapped on tori, and turn on magnetic fields on their
worldvolume. We then compare predictions for the spectrum of open strings
stretching between these D-branes, from perturbative string theory and from the
effective NBI action. Under some plausible assumptions, we find corrections to
the Str-prescription for the NBI action at order F^6. In the process we give a
way to classify terms in the NBI action that can be written in terms of field
strengths only, in terms of permutation group theory.Comment: LaTeX, 31 pages, 30 figure
Representations on the cohomology of hypersurfaces and mirror symmetry
We study the representation of a finite group acting on the cohomology of a
non-degenerate, invariant hypersurface of a projective toric variety. We deduce
an explicit description of the representation when the toric variety has at
worst quotient singularities. As an application, we conjecture a
representation-theoretic version of Batyrev and Borisov's mirror symmetry
between pairs of Calabi-Yau hypersurfaces, and prove it when the hypersurfaces
are both smooth or have dimension at most 3. An interesting consequence is the
existence of pairs of Calabi-Yau orbifolds whose Hodge diamonds are mirror,
with respect to the usual Hodge structure on singular cohomology.Comment: 34 page
Complete spectrum of the infinite- Hubbard ring using group theory
We present a full analytical solution of the multiconfigurational
strongly-correlated mixed-valence problem corresponding to the -Hubbard ring
filled with electrons, and infinite on-site repulsion. While the
eigenvalues and the eigenstates of the model are known already, analytical
determination of their degeneracy is presented here for the first time. The
full solution, including degeneracy count, is achieved for each spin
configuration by mapping the Hubbard model into a set of Huckel-annulene
problems for rings of variable size. The number and size of these effective
Huckel annulenes, both crucial to obtain Hubbard states and their degeneracy,
are determined by solving a well-known combinatorial enumeration problem, the
necklace problem for beads and two colors, within each subgroup of the
permutation group. Symmetry-adapted solution of the necklace
enumeration problem is finally achieved by means of the subduction of coset
representation technique [S. Fujita, Theor. Chim. Acta 76, 247 (1989)], which
provides a general and elegant strategy to solve the one-hole infinite-
Hubbard problem, including degeneracy count, for any ring size. The proposed
group theoretical strategy to solve the infinite- Hubbard problem for
electrons, is easily generalized to the case of arbitrary electron count ,
by analyzing the permutation group and all its subgroups.Comment: 31 pages, 4 figures. Submitte
Simple-Current Symmetries, Rank-Level Duality, and Linear Skein Relations for Chern-Simons Graphs
A previously proposed two-step algorithm for calculating the expectation
values of Chern-Simons graphs fails to determine certain crucial signs. The
step which involves calculating tetrahedra by solving certain non- linear
equations is repaired by introducing additional linear equations. As a first
step towards a new algorithm for general graphs we find useful linear equations
for those special graphs which support knots and links. Using the improved set
of equations for tetrahedra we examine the symmetries between tetrahedra
generated by arbitrary simple currents. Along the way we uncover the classical
origin of simple-current charges. The improved skein relations also lead to
exact identities between planar tetrahedra in level and level
CS theories, where denotes a classical group. These results are
recast as identities for quantum -symbols and WZW braid matrices. We obtain
the transformation properties of arbitrary graphs and links under simple
current symmetries and rank-level duality. For links with knotted components
this requires precise control of the braid eigenvalue permutation signs, which
we obtain from plethysm and an explicit expression for the (multiplicity free)
signs, valid for all compact gauge groups and all fusion products.Comment: 58 pages, BRX-TH-30
Finite Group Modular Data
In a remarkable variety of contexts appears the modular data associated to
finite groups. And yet, compared to the well-understood affine algebra modular
data, the general properties of this finite group modular data has been poorly
explored. In this paper we undergo such a study. We identify some senses in
which the finite group data is similar to, and different from, the affine data.
We also consider the data arising from a cohomological twist, and write down,
explicitly in terms of quantities associated directly with the finite group,
the modular S and T matrices for a general twist, for what appears to be the
first time in print.Comment: 38 pp, latex; 5 references added, "questions" section touched-u
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