A Note on the Orlik–Solomon Algebra

Let M = M(E) be a matroid on a linear ordered set E. The Orlik–Solomon Z -algebra OS(M) of M is the free exteriorZ -algebra on E, modulo the ideal generated by the circuit boundaries. The Z -moduleOS (M) has a canonical basis called ‘no broken circuit basis’ and denotednbc. Let eX = ∏ei,ei ∈ X ⊂ E. We prove that wheneX is expressed in thenbcbasis, then all the coefficients are 0 or ± 1. We present here an algorithm for computing these coefficients. We prove in appendix a numerical identity involving the dimensions of the algebras of Orlik–Solomon of the minors of a matroid and its dual.