Lexsegment and Gotzmann ideals associated with the diagonal action of \(\mathbb Z/p\)
Let \(R:=F[x_1,\dots,x_n]\) be a polynomial ring over a field \(F\). Let \(G\) be the cyclic group \(\mathbb Z/p\), \(p\) a prime. Consider the diagonal action of \(G\) on the polynomial ring \(R\). The ring of invariants \(R^G=\{f\in R\mid g(f)=f\;\forall g\in G\}\) is a finitely generated algebra. An important object in \(R\) is the Hilbert ideal \(H\) which is generated by the invariants of positive degree. Other important objects are lexsegment and Gotzmann ideals. These ideals have certain nice combinatorial properties and they form an important class of ideals in the study of Hilbert series of homogeneous ideals. A separating set is a subset of \(R^G\) and has the same separating power as the full ring of invariants. In this paper, the author studies the lexsegment and the Gotzmann properties of ideals generated by invariants of \(G\). He establishes a connection between the Gotzmann property of \(H\) and the action of the group \(G\). He develops criteria to detect the Gotzmann property for the case \(n=2\). Further, he shows that there is a separating set of monomials in \(R^G\) that generates a proper lexsegment ideal in \(R\) for all \(n\) and gives an algorithm to compute this set. He also presents the invariant ring corresponding to the diagonal action of \(\mathbb Z/2\times\mathbb Z/2\) that has no graded separating subalgebra whose elements of positive degree generate a Gotzmann ideal in \(R\).
scientific article; zbMATH DE number 5903708
scientific article