### 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 ℂ^{r}, A is isomorphic to the cohomology algebra of the complement ℂ^{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 ℂ^{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.

Original language | English (US) |
---|---|

Pages (from-to) | 189-199 |

Number of pages | 11 |

Journal | Journal of Algebraic Combinatorics |

Volume | 10 |

Issue number | 2 |

State | Published - Sep 1999 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Algebra and Number Theory
- Discrete Mathematics and Combinatorics

### Cite this

*Journal of Algebraic Combinatorics*,

*10*(2), 189-199.

**Orlik-Solomon Algebras and Tutte Polynomials.** / Eschenbrenner, Carrie J.; Falk, Michael J.

Research output: Contribution to journal › Article

*Journal of Algebraic Combinatorics*, vol. 10, no. 2, pp. 189-199.

}

TY - JOUR

T1 - Orlik-Solomon Algebras and Tutte Polynomials

AU - Eschenbrenner, Carrie J.

AU - Falk, Michael J

PY - 1999/9

Y1 - 1999/9

N2 - 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.

AB - 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.

KW - Arrangement

KW - Matroid

KW - Orlik-Solomon algebra

KW - Tutte polynomial

UR - http://www.scopus.com/inward/record.url?scp=0012915921&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0012915921&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0012915921

VL - 10

SP - 189

EP - 199

JO - Journal of Algebraic Combinatorics

JF - Journal of Algebraic Combinatorics

SN - 0925-9899

IS - 2

ER -