### Abstract

Let G be a matroid on ground set A. The Orlik-Solomon algebra A(G) is the quotient of the exterior algebra ℬ on A by the ideal ℐ generated by circuit boundaries. The quadratic closure A(G) of A(G) is the quotient of ℬ by the ideal generated by the degree-two component of ℐ. We introduce the notion of the nbb set in G, determined by a linear order on A, and show that the corresponding monomials are linearly independent in the quadratic closure A(G). As a consequence, A(G) is a quadratic algebra only if G is line-closed. An example of S. Yuzvinsky proves the converse false. [G. Denham and S. Yuzvinsky, Adv. in Appl. Math. 28, 2002, doi:10.1006/aama.2001.0779]. These results generalize to the degree r closure of A(G). The motivation for studying line-closed matroids grew out of the study of formal arrangements. This is a geometric condition necessary for A to be free and for the complement M of A to be a K(π, 1) space. Formality of A is also necessary for A(G) to be a quadratic algebra. We clarify the relationship between formality, line-closure, and other matroidal conditions related to formality. We give examples to show that line-closure of G is not necessary or sufficient for M to be a K(π, 1) or for A to be free.

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

Pages (from-to) | 250-271 |

Number of pages | 22 |

Journal | Advances in Applied Mathematics |

Volume | 28 |

Issue number | 2 |

DOIs | |

State | Published - 2002 |

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### ASJC Scopus subject areas

- Applied Mathematics
- Discrete Mathematics and Combinatorics

### Cite this

**Line-closed matroids, quadratic algebras, and formal arrangements.** / Falk, Michael J.

Research output: Contribution to journal › Article

*Advances in Applied Mathematics*, vol. 28, no. 2, pp. 250-271. https://doi.org/10.1006/aama.2001.0780

}

TY - JOUR

T1 - Line-closed matroids, quadratic algebras, and formal arrangements

AU - Falk, Michael J

PY - 2002

Y1 - 2002

N2 - Let G be a matroid on ground set A. The Orlik-Solomon algebra A(G) is the quotient of the exterior algebra ℬ on A by the ideal ℐ generated by circuit boundaries. The quadratic closure A(G) of A(G) is the quotient of ℬ by the ideal generated by the degree-two component of ℐ. We introduce the notion of the nbb set in G, determined by a linear order on A, and show that the corresponding monomials are linearly independent in the quadratic closure A(G). As a consequence, A(G) is a quadratic algebra only if G is line-closed. An example of S. Yuzvinsky proves the converse false. [G. Denham and S. Yuzvinsky, Adv. in Appl. Math. 28, 2002, doi:10.1006/aama.2001.0779]. These results generalize to the degree r closure of A(G). The motivation for studying line-closed matroids grew out of the study of formal arrangements. This is a geometric condition necessary for A to be free and for the complement M of A to be a K(π, 1) space. Formality of A is also necessary for A(G) to be a quadratic algebra. We clarify the relationship between formality, line-closure, and other matroidal conditions related to formality. We give examples to show that line-closure of G is not necessary or sufficient for M to be a K(π, 1) or for A to be free.

AB - Let G be a matroid on ground set A. The Orlik-Solomon algebra A(G) is the quotient of the exterior algebra ℬ on A by the ideal ℐ generated by circuit boundaries. The quadratic closure A(G) of A(G) is the quotient of ℬ by the ideal generated by the degree-two component of ℐ. We introduce the notion of the nbb set in G, determined by a linear order on A, and show that the corresponding monomials are linearly independent in the quadratic closure A(G). As a consequence, A(G) is a quadratic algebra only if G is line-closed. An example of S. Yuzvinsky proves the converse false. [G. Denham and S. Yuzvinsky, Adv. in Appl. Math. 28, 2002, doi:10.1006/aama.2001.0779]. These results generalize to the degree r closure of A(G). The motivation for studying line-closed matroids grew out of the study of formal arrangements. This is a geometric condition necessary for A to be free and for the complement M of A to be a K(π, 1) space. Formality of A is also necessary for A(G) to be a quadratic algebra. We clarify the relationship between formality, line-closure, and other matroidal conditions related to formality. We give examples to show that line-closure of G is not necessary or sufficient for M to be a K(π, 1) or for A to be free.

UR - http://www.scopus.com/inward/record.url?scp=0036228664&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036228664&partnerID=8YFLogxK

U2 - 10.1006/aama.2001.0780

DO - 10.1006/aama.2001.0780

M3 - Article

VL - 28

SP - 250

EP - 271

JO - Advances in Applied Mathematics

JF - Advances in Applied Mathematics

SN - 0196-8858

IS - 2

ER -