Action of the Borel group on monomial ideals

Let T < GL_(n,K) be the Borel group of upper triangular matrices. In this paper we want to study the action of T on the set of monomial ideals of K[x_1,...,x_n] (K a field of characteristic zero) from a computational point of view. More specifically, we show that the stabilizer of a monomial ideal M < K[x_1,...,x_n] in T is a purely combinatorial object and we give an algorithm for computing it. Then we characterize the subgroups of T that are stabilizers of monomial ideals, we give an algorithm which finds if a given ideal J is in the orbit of a monomial ideal M under the action of T and in the affirmative case, finds the matrices W in T such that W * J = M. We show that the entries of W can be directly obtained from the coefficients of the generators of J, so in particular no solutions of polynomial equations are required.