Eli Aljadeff, Alexei Kanel-Belov. Hilbert series of Pi relatively free G-graded algebras are rational functions

Natural Sciences / Mathematics / Algebra

Submitted on: Aug 27, 2012, 22:36:39

Description: 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 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 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/I is a rational function. More generally, we show that the Hilbert series which corresponds to any g-homogeneous component of F/I is a rational function.