Bases in Orlik-Solomon type algebras

Abstract

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 ℑ(M) be the ideal of E generated by the squares e12,..., e n2, elements of the form eiej + aijejei and 'boundaries of circuits', i.e., elements of the form ∑χjejl ... eij-1 eij+1 ... eim, with χj ε k and e il,..., eim a circuit of the matroid with some special coefficients. The χ-algebra A(M) is defined as the quotient of E by ℑ(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 ℑ(M) for any term order and give some consequences. © 2002 Elsevier Science Ltd. All rights reserved.