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Products of linear forms and Tutte polynomials. (English) Zbl 1219.05032
Summary: Let $$\Delta$$ be a finite sequence of $$n$$ vectors from a vector space over any field. We consider the subspace of $$\text{Sym}(V)$$ spanned by $$\prod _{v \in S}v$$, where $$S$$ is a subsequence of $$\Delta$$. A result of Orlik and Terao [P. Orlik and |it H. Terao, “Commutative algebras for arrangements,” Nagoya Math. J. 134, 65–73 (1994; Zbl 0801.05019)] provides a doubly indexed direct sum decomposition of this space. The main theorem is that the resulting Hilbert series is the Tutte polynomial evaluation $$T(\Delta ;1+x,y)$$. Results of Ardila and Postnikov [F. Ardila and |it A. Postnikov, “Combinatorics and geometry of power ideals,” Trans. Am. Math. Soc. 362, No. 8, 4357–4384 (2010; Zbl 1226.05019)], Orlik and Terao [loc. cit.], Terao [H. Terao, “Algebras generated by reciprocals of linear forms,” J. Algebra 250, No. 2, 549–558 (2002; Zbl 1049.13011)] , and Wagner [D.G. Wagner, “Algebras related to matroids represented in characteristic zero,” Eur. J. Comb. 20, No. 7, 701–711 (1999; Zbl 0996.16027)] are obtained as corollaries.

##### MSC:
 05B35 Combinatorial aspects of matroids and geometric lattices 52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
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##### References:
 [1] Federico Ardila, Enumerative and algebraic aspects of matroids and hyperplane arrangements, Ph.D. Thesis, Massachusetts Institute of Technology, February 2003. [2] Federico Ardila, Alexander Postnikov, Combinatorics and geometry of power ideals, Trans. Amer. Math. Soc. (2010) (in press). · Zbl 1226.05019 [3] Berget, Andrew, A short proof of gamas’s theorem, Linear algebra appl., 430, 2-3, 791-794, (2009) · Zbl 1183.15022 [4] Björner, Anders, The homology and shellability of matroids and geometric lattices, (), 226-283 · Zbl 0772.05027 [5] Brion, Michel; Vergne, Michèle, Arrangement of hyperplanes. I. rational functions and jeffrey – kirwan residue, Ann. sci. éc. norm. supér. (4), 32, 5, 715-741, (1999) · Zbl 0945.32003 [6] Brylawski, Thomas; Oxley, James, The Tutte polynomial and its applications, (), 123-225 · Zbl 0769.05026 [7] Cordovil, Raul, A commutative algebra for oriented matroids, Discrete comput. geom., 27, 1, 73-84, (2002), Geometric combinatorics (San Francisco, CA/Davis, CA, 2000) · Zbl 1016.52014 [8] Crapo, Henry H., The Tutte polynomial, Aequationes math., 3, 211-229, (1969) · Zbl 0197.50202 [9] Crapo, Henry; Schmitt, William, The Whitney algebra of a matroid, J. combin. theory ser. A, 91, 1-2, 215-263, (2000), In memory of Gian-Carlo Rota · Zbl 0964.05018 [10] De Concini, Corrado; Procesi, Claudio, Hyperplane arrangements and box splines, Michigan math. J., 57, 1-26, (2008), With an appendix by Anders Björner · Zbl 1217.14001 [11] Forge, David; Las Vergnas, Michel, Orlik – solomon type algebras, European J. combin., 22, 5, 699-704, (2001), Combinatorial geometries (Luminy, 1999) · Zbl 0984.52016 [12] Orlik, Peter; Terao, Hiroaki, Commutative algebras for arrangements, Nagoya math. J., 134, 65-73, (1994) · Zbl 0801.05019 [13] Postnikov, Alexander; Shapiro, Boris; Shapiro, Mikhail, Algebras of curvature forms on homogeneous manifolds (English summary), (), 227-235 · Zbl 0971.53034 [14] Proudfoot, Nicholas; Speyer, David, A broken circuit ring, Beiträge algebra geom., 47, 1, 161-166, (2006) · Zbl 1095.13024 [15] Hal Schenck, Stefan Tohaneanu, Orlik-Terao algebra and 2-formality, 2009, arXiv:0901.0253. [16] Terao, Hiroaki, Algebras generated by reciprocals of linear forms, J. algebra, 250, 2, 549-558, (2002) · Zbl 1049.13011 [17] Wagner, David G., Algebras related to matroids represented in characteristic zero, European J. combin., 20, 7, 701-711, (1999) · Zbl 0996.16027 [18] () [19] ()
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