Hilbert series of PI relatively free G-graded algebras are rational functions

TL;DR

This paper proves that the Hilbert series of G-graded relatively free PI algebras and their homogeneous components are always rational functions, extending understanding of their algebraic structure.

Contribution

It establishes the rationality of Hilbert series for G-graded PI algebras and their components, a new result in the theory of graded algebra identities.

Findings

Hilbert series of G-graded PI algebras are rational functions

The rationality extends to homogeneous components of these algebras

Provides a foundation for further algebraic and combinatorial analysis

Abstract

Let G be a finite group, (g_{1},...,g_{r}) an (unordered) r-tuple of G^{(r)} and x_{i,g_i}'s variables that correspond to the g_i's, i=1,...,r. Let F<x_{1,g_1},...,x_{r,g_r}> be the corresponding free G-graded algebra where F is a field of zero characteristic. Here the degree of a monomial is determined by the product of the indices in G. Let I be a G-graded T-ideal of F<x_{1,g_1},...,x_{r,g_r}> which is PI (e.g. any ideal of identities of a G-graded finite dimensional algebra is of this type). We prove that the Hilbert series of F<x_{1,g_1},...,x_{r,g_r}>/I is a rational function. More generally, we show that the Hilbert series which corresponds to any g-homogeneous component of F<x_{1,g_1},...,x_{r,g_r}>/I is a rational function.