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

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MR2966998 Reviewed Aljadeff, Eli(IL-TECH); Kanel-Belov, Alexei(IL-BILN) Hilbert series of PI relatively free $G$-graded algebras are rational functions. (English summary) Bull. Lond. Math. Soc. 44 (2012), no. 3, 520–532. 16R10 (16W50) Review PDF Clipboard Journal Article Make Link Let $F$ be a field of characteristic zero and $F\langle x_1, \ldots, x_n \rangle$ the free associative algebra on $x_1, \ldots, x_n$ over $F.$ In [Uspekhi Mat. Nauk 52 (1997), no. 2(314), 153–154; MR1480146] A. Kanel-Belov proved that if $\scr{I}$ is a $T$-ideal of $F\langle x_1, \ldots, x_n \rangle$, then the relatively free algebra $F\langle x_1, \ldots, x_n \rangle/\scr{I}$ has rational Hilbert series. Here the authors extend this result to the setting of group graded algebras as follows: let $G$ be a finite group and let $\scr{G}=(g_1, \ldots, g_r)$ be a fixed $r$-tuple of elements of $G$. Consider the free algebra $F\langle {\scr X_g} \rangle=F\langle x_{1,g_1}, \ldots, x_{r,g_r} \rangle$ on a set of elements indexed by the given $r$-tuple. $F\langle {\scr X_g} \rangle$ can be naturally graded by requiring that the homogeneous degree of a monomial $x_{s_1, g_{s_1}}\cdots x_{s_m, g_{s_m}}$ be $g_{s_1}\cdots g_{s_m}.$ Fix a $G$-graded $T$-ideal $\scr{I}$ of $F\langle {\scr X_g} \rangle$, i.e., an ideal invariant under all $G$-graded endomorphisms of $F\langle {\scr X_g}\rangle$. Let $\Omega_n$ be the finite set of monomials of degree $n$ on $\{x_{i, g_i}\}_{i=1}^r$ and let $c_n$ be the dimension of the $F$-subspace of $F\langle {\scr X_g} \rangle/\scr{I}$ spanned by the monomials of $\Omega_n$. Then $$ H_{F\langle {\scr X_g}\rangle/\scr{I}}(t)=\sum_n c_nt^n $$ is the Hilbert series of the relatively free $G$-graded algebra $F\langle {\scr X_g}\rangle/\scr{I}$ with respect to the generators represented by $\{x_{i, g_i}\}_{i=1}^r.$ In this paper the authors prove that, if $F\langle {\scr X_g} \rangle/\scr{I}$ is PI, i.e, it satisfies an ordinary polynomial identity, then the Hilbert series $F\langle {\scr X_g} \rangle/\scr{I}$ is the Taylor series of a rational function. More generally, they show that the Hilbert series which corresponds to any $g$-homogeneous component of $F\langle {\scr X_g}\rangle/\scr{I}$ is a rational function. Reviewed by Daniela La Mattina

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