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| Brion's theorem | |
|---|---|
| Name | Brion's theorem |
| Field | Combinatorics; Convex geometry; Algebraic geometry |
| Introduced | 1988 |
| Attributed to | Michel Brion |
| Keywords | polytope, lattice point, generating function |
Brion's theorem is a result about lattice-point generating functions of convex rational polytopes that expresses the global generating function as a finite sum of contributions from vertices. It links topics in Combinatorics, Convex geometry, Algebraic geometry, Toric variety, and Representation theory, and has influenced work in Ehrhart theory, Pick's theorem, Todd class, and computational integer programming.
For a convex rational polytope P in a real vector space V with lattice Λ, the lattice-point generating function of P is equal to the sum of lattice-point generating functions of the tangent cones at the vertices of P. The statement relates a global series for P to local series at each vertex, giving an equality in the ring of formal Laurent series; this links Ehrhart polynomial behavior to local data at each vertex and to the geometry of the associated fan of normal cones. The theorem applies to rational polytopes arising in contexts such as Birkhoff polytope, permutahedron, associahedron, Gelfand–Zetlin polytope, and other polytopes tied to Lie group and Weyl group combinatorics.
The theorem arose in the late 1980s in interactions among researchers studying lattice points in polytopes, toric geometry, and representation theory, connecting work of Ehrhart, Pick, Pommersheim, Khovanskii, Khovanskii–Pukhlikov, Stanley, and Atiyah. It synthesizes classical results like Ehrhart polynomial enumeration and modern tools from Barvinok-style generating function algorithms and Todd class computations on toric varietys associated to rational fans. Applications trace to problems studied by authors working on Riemann–Roch theorem for toric varieties, combinatorial formulas from Schubert calculus, and algorithmic lattice-point enumeration in the traditions of Lenstra and Kannan–Lovász.
One approach constructs the equality by decomposing the indicator function of P into a finite alternating sum of indicator functions of tangent cones and then passing to generating functions; this route uses results akin to those used by Guillemin and Sternberg in equivariant index theory and by Beck and Robins in lattice-point enumeration. An alternative proof uses the geometry of toric varieties, interpreting the generating series as characters of graded rings and invoking the Riemann–Roch theorem or calculation of Todd class on the toric variety associated to the normal fan of P, following ideas connected to Danilov and Khovanskii. A combinatorial analytic proof employs Fourier analysis on the lattice and rational function identities similar to techniques in work by Barvinok and Baldoni-Silva.
Brion-style decompositions underpin algorithms for computing lattice-point enumerators in fixed dimension as in Barvinok's algorithm, and they inform symbolic formulas used in Ehrhart theory and the computation of Ehrhart polynomial coefficients. In Representation theory, the theorem aids in counting weight multiplicities via polytopes such as Gelfand–Zetlin polytope and Kostant's partition function domains related to Weyl group combinatorics and Kostant's multiplicity formula. In Algebraic geometry, the local-to-global description contributes to calculations in toric variety cohomology, intersection theory relating to Todd class and Hirzebruch–Riemann–Roch theorem, and to formulae in Schubert calculus for flag variety degenerations. Computationally, it features in integer optimization techniques that build on Lenstra and Kannan frameworks and in software developed for SageMath, LattE, and related projects.
For simple rational polytopes like a unimodular simplex, the generating function equals a single rational function coming from the unique vertex, illustrating connections to Pick's theorem in low dimensions. For the Birkhoff polytope and the permutahedron, Brion-type decompositions express counting formulas for magic squares and permutation-related lattice points, tying to work by Stanley on order polytopes and Postnikov on generalized permutahedra. In the case of the Gelfand–Zetlin polytope, the vertex-cone contributions relate directly to multiplicity formulas appearing in Weyl character formula analyses and to combinatorial models studied by Knutson and Tao.
Generalizations include analogues for polyhedral complexes, equivariant versions in the context of Hamiltonian Guillemin–Sternberg convexity, and refinements that track coefficients in graded rings appearing in Riemann–Roch type formulae; these extend connections to Poincaré duality phenomena in toric contexts. Related results encompass Ehrhart–Macdonald reciprocity, Khovanskii–Pukhlikov formula, and computational refinements in the line of Barvinok and De Loera that yield efficient enumeration in fixed dimension. The theorem also interfaces with ongoing research on discrete geometry problems studied by communities around Discrete Geometry conferences and programs at institutions such as MSRI, ICMS, and research groups linked to Institut des Hautes Études Scientifiques.
Category:Theorems in combinatorics