an:05801176
Zbl 1219.05032
Berget, Andrew
Products of linear forms and Tutte polynomials
EN
Eur. J. Comb. 31, No. 7, 1924-1935 (2010).
00267000
2010
j
05B35 52B05
direct sum decomposition; Hilbert series; Tutte polynomial evaluation; square free monomials
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 [\textit{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 [\textit{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 [\textit{H. Terao}, ``Algebras generated by reciprocals of linear forms,'' J. Algebra 250, No.\,2, 549--558 (2002; Zbl 1049.13011)] , and Wagner [\textit{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.
Zbl 0801.05019; Zbl 1049.13011; Zbl 0996.16027; Zbl 1226.05019