Bases in Orlik–Solomon Type Algebras

Let M be a matroid on [ n ] and E be the graded algebra generated over a field k generated by the elements 1, e1,⋯ , en. Let (M) be the ideal of E generated by the squares e12,⋯ , en2, elements of the form eiej + aijejeiand ‘boundaries of circuits’, i.e., elements of the form ∑χjei1⋯eij − 1eij + 1⋯eim, with χj ∈ k and ei1,⋯ , eima circuit of the matroid with some special coefficients. The χ -algebra A(M) is defined as the quotient of E by (M). Recall that the class of χ -algebras contains several studied algebras and in first place the Orlik–Solomon algebra of a matroid. We will essentially construct the reduced Gröbner basis of (M) for any term order and give some consequences.