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 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 |
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Keywords
- Arrangement
- Matroid
- Orlik-Solomon algebra
- Tutte polynomial
ASJC Scopus subject areas
- Mathematics(all)
- Algebra and Number Theory
- Discrete Mathematics and Combinatorics
Cite this
Orlik-Solomon Algebras and Tutte Polynomials. / Eschenbrenner, Carrie J.; Falk, Michael J.
In: Journal of Algebraic Combinatorics, Vol. 10, No. 2, 09.1999, p. 189-199.Research output: Contribution to journal › Article
}
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 -