Volume
and Number of Integral Points in Rational Polyhedra

This webpage hosts papers and softwares about the problem of computing the volume and the number of integral points in rational polyhedra (and its applications to representation theory).

- Volume computation for
polytopes and partition functions for classical root systems.

Authors: M. W. Baldoni, M. Beck, C. Cochet, M. Vergne.

Abstract: 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.

Paper: ps (2,3 Mo), pdf (500 ko).

Status of the paper: submitted. Also available on the arXiv:math.CO/0504231.

Softwares: compressed archive (60 ko) containing Maple worksheets.

- Vector Partition and
Representation Theory

Author: C. Cochet.

Abstract: We apply some recent developments of Baldoni-Beck-Cochet-Vergne on vector partition function, to Kostant's and Steinberg's formulae, for classical Lie algebras $A_r$, $B_r$, $C_r$, $D_r$. We therefore get efficient {\tt Maple} programs that compute for these Lie algebras: the multiplicity of a weight in an irreducible finite-dimensional representation; the decomposition coefficients of the tensor product of two irreducible finite-dimensional representations. These programs can also calculate associated Ehrhart quasipolynomials.

Paper: ps (2,6 Mo), pdf (500 ko).

Status of the paper: accepted for the proceedings of the conference Formal Power Series and Algebraic Combinatorics 2005 (June 20-25, Taormina, Italy), being reviewed.

Softwares: not yet distributed.

- Counting Integer Flows in
Networks

Authors: M. W. Baldoni, J. De Loera, M. Vergne.

Abstract: This paper discusses new analytic algorithms and software for the enumeration of all integer flows inside a network. Concrete applications abound in graph theory \cite{Jaeger}, representation theory \cite{kirillov}, and statistics \cite{persi}. Our methods clearly surpass traditional exhaustive enumeration and other algorithms and can even yield formulas when the input data contains some parameters. These methods are based on the study of rational functions with poles on arrangements of hyperplanes.

Paper: ps (500 ko), pdf (400 ko).

Status of the paper: Found. Comput. Math. 4 (2004), no.3, 277-314. Also available on the arXiv:math.CO/0303228.

Softwares: compressed archive (12 ko) containing Maple worksheets.

- Title of the paper

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