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Orlik-Solomon algebras and Tutte polynomials. (English) Zbl 0955.52010
Summary: The \(OS\) algebra \(A\) of a matroid \(M\) is a graded algebra related to the Whitney homology of the lattice of flats of \(M\). In case \(M\) is the underlying matroid of a hyperplane arrangement \({\mathcal A}\) in \(\mathbb{C}^r \), \(A\) is isomorphic to the cohomology algebra of the complement \(\mathbb{C}^r \setminus \bigcup{\mathcal A}\). Few examples are known of pairs of arrangements with non-isomorphic matroids but isomorphic \(OS\) algebras. In all known examples, the Tutte polynomials are identical, and the complements are homotopy equivalent but not homeomorphic.
We construct, for any given simple matroid \(M_0\), a pair of infinite families of matroids \(M_n\) and \(M_n'\), \(n\geq 1\), each containing \(M_0\) as a submatroid, in which corresponding pairs have isomorphic \(OS\) algebras. If the seed matroid \(M_0\) is connected, then \(M_n\) and \(M_n'\) have different Tutte polynomials. As a consequence of the construction, we obtain, for any \(m,m\) different matroids with isomorphic \(OS\) algebras. Suppose one is given a pair of central complex hyperplane arrangements \({\mathcal A}_0\) and \({\mathcal A}_1\). Let \({\mathcal S}\) denote the arrangement consisting of the hyperplane \(\{0\}\) in \(\mathbb{C}^1\). We define the parallel connection \(P({\mathcal A}_0,{\mathcal A}_1)\), an arrangement realizing the parallel connection of the underlying matroids, and show that the direct sums \({\mathcal A}_0\oplus {\mathcal A}_1\) and \({\mathcal S}\oplus P({\mathcal A}_0, {\mathcal A}_1)\) have diffeomorphic complements.

52B40 Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.)
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
05B35 Combinatorial aspects of matroids and geometric lattices
Full Text: DOI arXiv
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