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

Eli Aljadeff; Alexei Kanel-Belov

doi:10.1112/blms/bdr116

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

Eli Aljadeff, Alexei Kanel-Belov

Department of Mathematics

Technion-Israel Institute of Technology

Research output: Contribution to journal › Article › peer-review

12 Scopus citations

Abstract

Given a finite group G and a field F of characteristic zero, we let F〈x₁,g₁,⋯,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〈x₁,g₁,⋯,x_r,g _r〉 which is PI (that is, the algebra F〈x₁,g ₁,⋯,x_r,g_r〉/ℐ is PI). We prove that the Hilbert series of F〈₁,g₁,⋯,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,g₁,⋯,x_r,g _r〉/ℐ is a rational function.

Original language	English
Pages (from-to)	520-532
Number of pages	13
Journal	Bulletin of the London Mathematical Society
Volume	44
Issue number	3
DOIs	https://doi.org/10.1112/blms/bdr116
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
E. Schaver Research Fund	1178/06
Russian Fund of Fundamental Research
Israel Science Foundation	1283/08

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10.1112/blms/bdr116

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title = "Hilbert series of PI relatively free G-graded algebras are rational functions",

abstract = "Given a finite group G and a field F of characteristic zero, we let F〈x1,g1,⋯,xr,gr〉 be the free G-graded F-algebra generated by homogeneous variables {x i,gi}gi∈ G. Let ℐ be a G-graded T-ideal of F〈x1,g1,⋯,xr,g r〉 which is PI (that is, the algebra F〈x1,g 1,⋯,xr,gr〉/ℐ is PI). We prove that the Hilbert series of F〈1,g1,⋯,x r,gr〉/ℐ is a rational function. More generally, we show that the Hilbert series which corresponds to any g-homogeneous component of F〈1,g1,⋯,xr,g r〉/ℐ is a rational function.",

author = "Eli Aljadeff and Alexei Kanel-Belov",

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).",

year = "2012",

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AU - Kanel-Belov, Alexei

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PY - 2012/6

Y1 - 2012/6

N2 - Given a finite group G and a field F of characteristic zero, we let F〈x1,g1,⋯,xr,gr〉 be the free G-graded F-algebra generated by homogeneous variables {x i,gi}gi∈ G. Let ℐ be a G-graded T-ideal of F〈x1,g1,⋯,xr,g r〉 which is PI (that is, the algebra F〈x1,g 1,⋯,xr,gr〉/ℐ is PI). We prove that the Hilbert series of F〈1,g1,⋯,x r,gr〉/ℐ is a rational function. More generally, we show that the Hilbert series which corresponds to any g-homogeneous component of F〈1,g1,⋯,xr,g r〉/ℐ is a rational function.

AB - Given a finite group G and a field F of characteristic zero, we let F〈x1,g1,⋯,xr,gr〉 be the free G-graded F-algebra generated by homogeneous variables {x i,gi}gi∈ G. Let ℐ be a G-graded T-ideal of F〈x1,g1,⋯,xr,g r〉 which is PI (that is, the algebra F〈x1,g 1,⋯,xr,gr〉/ℐ is PI). We prove that the Hilbert series of F〈1,g1,⋯,x r,gr〉/ℐ is a rational function. More generally, we show that the Hilbert series which corresponds to any g-homogeneous component of F〈1,g1,⋯,xr,g r〉/ℐ is a rational function.

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

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