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Volume computation for polytopes and partition functions for classical root systems. (English) Zbl 1105.52001
This paper develops an algorithm for computing the volume and number of integer points of a polytope. It is used to give a fast algorithm for the computation of the partition function of classical root systems. The approach is based on the inverse Laplace transform of rational functions on the complement of a hyperplane arrangement.
Jeffrey-Kirwan residues [L. C. Jeffrey and F. C. Kirwan, Topology 34, 291–327 (1995; Zbl 0833.55009)] are computed using the maximal nested sets of C. DeConcini and C. Procesi [Progress in Mathematics 235, 139–149 (2005; Zbl 1093.52503)]. The authors discuss implementation for the various classical root systems, and compare their algorithm with the Sp (special permutations) algorithm of Baldoni, et al., and with LattE, the software package developed by DeLoera, et al. (based on Barvinok’s algorithm).

52-04 Software, source code, etc. for problems pertaining to convex and discrete geometry
05E15 Combinatorial aspects of groups and algebras (MSC2010)
11P21 Lattice points in specified regions
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
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