Hilbert series of PI relatively free \(G\)-graded algebras are rational functions.
Let \(F\) be a field of characteristic 0 and let \(G\) be a finite group. Consider the free associative \(G\)-graded algebra \(F(X;G)\) on \(r\) homogeneous (in the \(G\)-grading) free generators. Take an ideal \(I\) in \(F(X;G)\) that is invariant under all endomorphisms that respect the \(G\)-grading. Such ideals are called graded T-ideals; these are the ideals of \(G\)-graded identities of a \(G\)-graded algebra. Suppose further that the relatively free graded algebra \(F(X;G)/I\) satisfies an ordinary polynomial identity.NEWLINENEWLINE The main theorem of the paper under review proves that the Hilbert series (in one variable) of \(F(X;G)/I\) is a rational function. Moreover the authors prove that the same holds if one considers the multivariate Hilbert series of \(F(X;G)/I\). Another result worth mentioning is that the Hilbert series of each \(G\)-homogeneous component of \(F(X;G)/I\) is also a rational function.NEWLINENEWLINE The proofs of these results are quite involved and will be of significant interest to the specialists in PI algebras.
scientific article; zbMATH DE number 6048037