Orlik-Solomon algebras and Tutte polynomials

Eschenbrenner, Carrie J. ; Falk, Michael J.

Journal of Algebraic Combinatorics, Tome 10 (1999) no. 2, pp. 189-199.

Résumé

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 $A$ in $Copf^{ r } , A$ is isomorphic to the cohomology algebra of the complement $Copf^{ r }setmncupA$. 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 $M \_{ n } ^$ prime$ , nge$ 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 $M \_{ n } ^$ prime 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 $cup _{1}$ . 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 $A_{0} oplusA _{1}$ and $SoplusP ( A _{0}, A _{1})$ have diffeomorphic complements.

Détail

Keywords: matroid, arrangement, orlik-Solomon algebra, tutte polynomial