Orlik-Solomon Algebras and Tutte Polynomials
Carrie J. Eschenbrenner
and Michael J. Falk
DOI: 10.1023/A:1018735815621
Abstract
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 A in 
r , A is isomorphic to the cohomology algebra of the complement r 
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.
r , A is isomorphic to the cohomology algebra of the complement r 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 
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 A 0 and A 1 . Let S denote the arrangement consisting of the hyperplane {0} in 1 . We define the parallel connection P( A 0, A 1), an arrangement realizing the parallel connection of the underlying matroids, and show that the direct sums A 0 
A 1 and S P ( A 0, A 1) have diffeomorphic complements.
, n 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 A 0 and A 1 . Let S denote the arrangement consisting of the hyperplane {0} in 1 . We define the parallel connection P( A 0, A 1), an arrangement realizing the parallel connection of the underlying matroids, and show that the direct sums A 0 A 1 and S P ( A 0, A 1) have diffeomorphic complements.Pages: 189–199
Keywords: matroid; arrangement; orlik-Solomon algebra; tutte polynomial
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References
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3. T.H. Brylawski, “A combinatorial model for series-parallel networks,” Transactions of the American Mathematical Society 154 (1971), 1-22.
4. M. Falk, “On the algebra associated with a geometric lattice,” Advances in Mathematics 80 (1989), 152-163.
5. M. Falk, “Homotopy, types of line arrangements,” Inventiones Mathematicae 111 (1993), 139-150.
6. M. Falk, “Arrangements and cohomology,” Annals of Combinatorics 1 (1997), 135-157.
7. Tan Jiang and Stephen S.-T. Yau, “Topological invariance of intersection lattices of arrangements in CP2,” Bulletin of the American Mathematical Society 29 (1993), 88-93.
8. P. Orlik and L. Solomon, “Topology and combinatorics of complements of hyperplanes,” Inventiones Mathematicae 56 (1980), 167-189.
9. P. Orlik and H. Terao, Arrangements of Hyperplanes, Springer Verlag, Berlin, Heidelberg, New York, 1992.
10. J. Oxley, Matroid Theory, Oxford University Press, Oxford, New York, Tokyo, 1992.
11. N. White (Ed.), Theory of Matroids, Cambridge University Press, Cambridge, 1986.
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