Line-Closed Matroids, Quadratic Algebras, and Formal Arrangments

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By this author: Falk M.

Author: Falk M.

Source: Advances in Applied Mathematics, Volume 28, Number 2, February 2002 , pp. 250-271(22)

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Abstract:

Let G be a matroid on ground set Ascr . The Orlik–Solomon algebra A(G) is the quotient of the exterior algebra Escr on Ascr by the ideal Iscr generated by circuit boundaries. The quadratic closure (G) of A(G) is the quotient of Escr by the ideal generated by the degree-two component of Iscr . We introduce the notion of the nbb set in G, determined by a linear order on Ascr , and show that the corresponding monomials are linearly independent in the quadratic closure (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 Ascr (G).

The motivation for studying line-closed matroids grew out of the study of formal arrangements. This is a geometric condition necessary for Ascr to be free and for the complement M of Ascr to be a K(, 1) space. Formality of Ascr 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(, 1) or for Ascr to be free. © 2002 Elsevier Science (USA).