1,328 research outputs found
Volume computation for polytopes and partition functions for classical root systems
This paper presents an algorithm to compute the value of the inverse Laplace
transforms of rational functions with poles on arrangements of hyperplanes. As
an application, we present an efficient computation of the partition function
for classical root systems.Comment: 55 pages, 14 figures. Maple programs available at
http://www.math.polytechnique.fr/~vergne/work/IntegralPoints.htm
On the Computation of Clebsch-Gordan Coefficients and the Dilation Effect
We investigate the problem of computing tensor product multiplicities for
complex semisimple Lie algebras. Even though computing these numbers is #P-hard
in general, we show that if the rank of the Lie algebra is assumed fixed, then
there is a polynomial time algorithm, based on counting the lattice points in
polytopes. In fact, for Lie algebras of type A_r, there is an algorithm, based
on the ellipsoid algorithm, to decide when the coefficients are nonzero in
polynomial time for arbitrary rank. Our experiments show that the lattice point
algorithm is superior in practice to the standard techniques for computing
multiplicities when the weights have large entries but small rank. Using an
implementation of this algorithm, we provide experimental evidence for
conjectured generalizations of the saturation property of
Littlewood--Richardson coefficients. One of these conjectures seems to be valid
for types B_n, C_n, and D_n.Comment: 21 pages, 6 table
Flow polytopes of signed graphs and the Kostant partition function
We establish the relationship between volumes of flow polytopes associated to
signed graphs and the Kostant partition function. A special case of this
relationship, namely, when the graphs are signless, has been studied in detail
by Baldoni and Vergne using techniques of residues. In contrast with their
approach, we provide entirely combinatorial proofs inspired by the work of
Postnikov and Stanley on flow polytopes. As a fascinating special family of
flow polytopes, we study the Chan-Robbins-Yuen polytopes. Motivated by the
beautiful volume formula for the type version,
where is the th Catalan number, we introduce type and
Chan-Robbins-Yuen polytopes along with intriguing conjectures
pertaining to their properties.Comment: 29 pages, 13 figure
Enumerative properties of generalized associahedra
Some enumerative aspects of the fans, called generalized associahedra,
introduced by S. Fomin and A. Zelevinsky in their theory of cluster algebras
are considered, in relation with a bicomplex and its two spectral sequences. A
precise enumerative relation with the lattices of generalized noncrossing
partitions is conjectured and some evidence is given.Comment: 15 page
New Strings for Old Veneziano Amplitudes III. Symplectic Treatment
A d-dimensional rational polytope P is a polytope whose vertices are located
at the nodes of d-dimensional Z-lattice. Consider a number of points inside the
inflated polytope (with coefficient of inflation k, k=1,2, 3...). The Ehrhart
polynomial of P counts the number of such lattice points (nodes) inside the
inflated P and (may be) at its faces (including vertices). In Part I
(hep-th/0410242) of our four parts work we noticed that the Veneziano amplitude
is just the Laplace transform of the generating function (considered as a
partition function in the sence of statistical mechanics) for the Ehrhart
polynomial for the regular inflated simplex obtained as a deformation retract
of the Fermat (hyper) surface living in complex projective space. This
observation is sufficient for development of new symplectic (this work) and
supersymmetric (hep-th/0411241)physical models reproducing the Veneziano (and
Veneziano-like) amplitudes. General ideas (e.g.those related to the properties
of Ehrhart polynomials) are illustrated by simple practical examples (e.g. use
of mirror symmetry for explanation of available experimental data on pion-pion
scattering) worked out in some detail. Obtained final results are in formal
accord with those earlier obtained by Vergne [PNAS 93 (1996) 14238].Comment: 48 pages J.Geom.Phys.(in press, available on line
Plethysm and lattice point counting
We apply lattice point counting methods to compute the multiplicities in the
plethysm of . Our approach gives insight into the asymptotic growth of
the plethysm and makes the problem amenable to computer algebra. We prove an
old conjecture of Howe on the leading term of plethysm. For any partition
of 3,4, or 5 we obtain an explicit formula in and for the
multiplicity of in .Comment: 25 pages including appendix, 1 figure, computational results and code
available at http://thomas-kahle.de/plethysm.html, v2: various improvements,
v3: final version appeared in JFoC
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