## Abstract

Given a finite group G and a field F of characteristic zero, we let F〈x_{1},g_{1},⋯,x_{r},g_{r}〉 be the free G-graded F-algebra generated by homogeneous variables {x _{i},g_{i}}g_{i}∈ G. Let ℐ be a G-graded T-ideal of F〈x_{1},g_{1},⋯,x_{r},g _{r}〉 which is PI (that is, the algebra F〈x_{1},g _{1},⋯,x_{r},g_{r}〉/ℐ is PI). We prove that the Hilbert series of F〈_{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〈_{1},g_{1},⋯,x_{r},g _{r}〉/ℐ is a rational function.

Original language | English |
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Pages (from-to) | 520-532 |

Number of pages | 13 |

Journal | Bulletin of the London Mathematical Society |

Volume | 44 |

Issue number | 3 |

DOIs | |

State | Published - Jun 2012 |

### Bibliographical note

Funding Information:The first author was partially supported by the Israel Science Foundation (grant no. 1283/08) and by the E. Schaver Research Fund. The second author was partially supported by the Israel Science Foundation (grant No. 1178/06). The second author is grateful to the Russian Fund of Fundamental Research for supporting his visit to India in 2008 (grant no. FBR 08-01-91300-INDa).

### Funding

The first author was partially supported by the Israel Science Foundation (grant no. 1283/08) and by the E. Schaver Research Fund. The second author was partially supported by the Israel Science Foundation (grant No. 1178/06). The second author is grateful to the Russian Fund of Fundamental Research for supporting his visit to India in 2008 (grant no. FBR 08-01-91300-INDa).

Funders | Funder number |
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E. Schaver Research Fund | 1178/06 |

Russian Fund of Fundamental Research | |

Israel Science Foundation | 1283/08 |