Orlik-Solomon Algebras and Tutte Polynomials

Carrie J. Eschenbrenner, Michael J. Falk

Research output: Contribution to journalArticle

15 Scopus citations

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 ℂr, A is isomorphic to the cohomology algebra of the complement ℂr \∪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 M0, a pair of infinite families of matroids Mn and M′n, n ≥ 1, each containing M0 as a submatroid, in which corresponding pairs have isomorphic OS algebras. If the seed matroid M0 is connected, then Mn 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 A0 and A1. Let S denote the arrangement consisting of the hyperplane {0} in ℂ1. We define the parallel connection P(A0, A1), an arrangement realizing the parallel connection of the underlying matroids, and show that the direct sums A0 ⊕ A1 and S ⊕ P (A0, A1) have diffeomorphic complements.

Original languageEnglish (US)
Pages (from-to)189-199
Number of pages11
JournalJournal of Algebraic Combinatorics
Volume10
Issue number2
DOIs
StatePublished - Jan 1 1999

Keywords

  • Arrangement
  • Matroid
  • Orlik-Solomon algebra
  • Tutte polynomial

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Discrete Mathematics and Combinatorics

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