Characterization of Hilbert functions of Gorenstein Artin algebras with the weak Stanley property

Abstract

We give a characterization of Hubert functions of Gorenstein Artin algebras with the weak Stanley property.Namely, we prove that a necessary and sufficient condition for a given O-sequence H_ to be the Hubert function of some Gorenstein Artin algebra with the weak Stanley property is that the sequence H is an Si-sequence. IntroductionA standard G-algebra A = A0 ® Ax © • • • over a field k is a commutative graded ring such that Aq = k, A is generated as a fc-algebra by Ax and the dimension of Ax as a /c-vector space is finite.Thus they are those A;-algebras which can be written in the form A = k[xo, xx, ... , xn]/I, where the jc; are indeterminates of degree 1 and / is a homogeneous ideal.In this paper a standard G-algebra over k is called a k-algebra for short.The Hubert function of A is defined by H(A, i) = dim^ A¡ for all / = 0, 1,... , and H(A, i) = 0 for all i < 0. In particular, if A is Artin, then we denote by the finite sequence H(A) = (H(A, 0), H(A, I), ... , H(A, s)) the Hubert function of A, where s = max{/ | A, ^ 0} .Furthermore, we put o(A) = s + 1.If h and i are positive integers, then h can be written uniquely in the form * = (?) + (?L-l) + -(7)where «, > w,-i > • • • > n¡ > j > 1.We put 0</> = 0 and »--(7++ó+C"-,+1)+---+(7++.)-A finite or infinite sequence H= (ho, hx, ...) of nonnegative integers is called an O-sequence if H is the Hubert function of some /c-algebra, i.e., Ao = 1 and 0 < Al+, < hfl> for i = 1,2,... (cf.[7] and [11, Theorem 2.2] for the details).A sequence H = (ho, hx, ... , hs) of positive integers is called a