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Kostant partitions functions and flow polytopes. (English) Zbl 1200.52008
Summary: This paper discusses volumes and Ehrhart polynomials in the context of flow polytopes. The general approach that studies these functions via rational functions with poles on an arrangement of hyperplanes and the total residue of such functions allows us, via a unified approach, to reobtain many interesting calculations existing in the literature. In particular we generalize Lidskii formula relating the Ehrhart polynomial to the volume function.

MSC:
52B12 Special polytopes (linear programming, centrally symmetric, etc.)
05A17 Combinatorial aspects of partitions of integers
11D45 Counting solutions of Diophantine equations
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
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[1] W. Baldoni, M. Beck, C. Cochet, M. Vergne, Volume computation for polytopes and partition functions for classical root systems, Discrete Comput. Geom. 35 (2006), 551–595. Programs available at www.math.polytechnique.fr/cmat/vergne/ . · Zbl 1105.52001
[2] W. Baldoni, J. de Loera, M. Vergne, Counting integer ows in networks, Found. Comput. Math. 4 (2004), 277–314. Programs available at www.math.ucdavis.edu/\(\sim\)stotalresidue/ · Zbl 1083.68640
[3] W. Baldoni-Silva, M. Vergne, Morris identities and the total residue for a system of type A r , in: Non-Commutative Harmonic Analysis, Progress in Mathematics, Vol. 220, Birkhäuser, Boston, 2004, pp. 1–19. · Zbl 1097.33011
[4] W. Baldoni-Silva, M. Vergne, Residues formulae for volumes and Ehrhart polynomials of convex polytopes (2001), 81 pp. Available at math.ArXiv,CO/0103097 .
[5] M. Brion, M. Vergne, Residue formulae, vector partition functions and lattice points in rational polytopes, J. Amer. Math Soc. 10 (1997), 797–833. · Zbl 0926.52016
[6] M. Brion, M. Vergne, Arrangements of hyperplanes I: Rational functions and Jeffrey–Kirwan residue, Ann. Sci. Ècole. Norm. Sup. 32 (1999), 715–741. · Zbl 0945.32003
[7] C. S. Chan, D. P. Robbins, D. S. Yuen, On the volume of a certain polytope, Experiment. Math. 9 (2000), 91–99. · Zbl 0960.05004
[8] C. Cochet, Vector partition function and representation theory (2005). Available at arXiv:math/0506159 . Programs available at www.institut.math.jussieu.fr/cochet/polytopes .
[9] C. Cochet, Kostka numbers and Littlewood–Richardson coefficients, in: Proceedings of AMS–IMS–SIAM Joint Summer Research Conference on Integer Points (Snowbird, UT, USA) 2003, Contemporary Mathematics, Vol. 374, American Mathematical Society, Providence, RI, 2005, pp. 79–89. Programs available at www.institut.math.jussieu.fr/cochet/polytopes .
[10] C. Cochet, Programme Maple calculant les chambres combinatoires pour la fonction de partition vectorielle. Available at www.institut.math.jussieu.fr/cochet/polytopes/Ichambers.txt .
[11] W. Dahmen, C. Micchelli, The number of solutions to linear Diophantine equations and multivariate splines, Trans. Amer. Math. Soc. 308 (1988), 509–532. · Zbl 0655.10013
[12] C. De Concini, C. Procesi, Nested sets and Jeffrey–Kirwan residues, in: Geometric Methods in Algebra and Number Theory, Progress in Mathematrics, Vol. 235, Birkhäuser, Boston, 2005, pp. 139–149. · Zbl 1093.52503
[13] C. De Concini, C. Procesi, Topics in hyperplane arrangements, polytopes and box splines, Preliminary version (October 2007). Available on www.mat.uniroma1.it/procesi/didattica . · Zbl 1217.14001
[14] L. C. Jeffrey, F. C. Kirwan, Localization for nonabelian group actions, Topology 34 (1995), 291–327. · Zbl 0833.55009
[15] А. Г. Хованский, А. В. Пухликов, Теорема Римана-Роха для интегралов и сумм квазиполиномов над виртуальными многогранниками, Алгебра и анализ 4 (1992), no. 4, 188–216. Engl. transl.: A. G. Khovanskii, A. V. Pukhlikov, A Riemann–Roch theorem for integrals and sums of quasipolynomials over virtual polytopes, St. Petersburg Math. J. 4 (1993), 789–812. · Zbl 0308.02034
[16] A. Kirillov, Ubiquity of Kostka polynomials, in: Physics and Combinatorics, 1999 (Nagoya), World Scientific, River Edge, NJ, 2001, pp. 85–200.
[17] G. I. Lehrer, On the Poincarè series associated with Coxeter group actions on complement of hyperplanes, J. London Math. Soc. 36 (1987), 275–294. · Zbl 0649.20041
[18] Б. В. Лидский, Фуккция Костанта системы корней A n , Функц. анал. и его прил. 18 (1984), no. 1, 76–77. Engl. transl.: B. V. Lidskij, Kostant function of the root system A n , Funct. Anal. Appl. 18 (1984), 65–67. · Zbl 0297.15002
[19] W. Morris, Constant term identities for finite and affine root systems (1982), PhD thesis, University of Wisconsin, Madison, WI.
[20] P.-E. Paradan, Jump formulas in Hamiltonian geometry (2004). Available at arXiv:math/0411306 .
[21] J. Pitman, R. P. Stanley, A polytope related to empirical distributions, plane trees, parking functions, and the associahedron, Discrete Comput. Geom. 27 (2002), 603–634. · Zbl 1012.52019
[22] J. R. Schmidt. A. M. Bincer, The Kostant partition function for simple Lie algebras, J. Math. Phys. 25 (1984), 2367–2373. · Zbl 0551.17002
[23] R. P. Stanley, Acyclic ow polytopes and Kostant partition function, Conference transparencies ( www-math.mit.edu/rstan/trans/htlm ).
[24] A. Szenes, M. Vergne, Residue formulae for vector partitions and Euler–MacLaurin sums, Adv. in Appl. Math. 30 (2003), 295–342. · Zbl 1067.52014
[25] D. Zeilberger, A conjecture of Chan, Robbins, and Yuen (1998). Available at math.ArXiv, CO/9811108 .
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