A Note on the Orlik-Solomon Algebra

LetM =M(E) be a matroid on a linear ordered set E . The Orlik–Solomon Z-algebra OS(M) of M is the free exterior Z-algebra on E , modulo the ideal generated by the circuit boundaries. The Z-module OS(M) has a canonical basis called ‘no broken circuit basis’ and denoted nbc. Let eX = ∏ ei , ei ∈ X ⊂ E . We prove that when eX is expressed in the nbc basis, 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.