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

Technion - Israel Institute of Technology, Bar-Ilan University

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

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

All Science Journal Classification (ASJC) codes

General Mathematics

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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",

month = jun,

doi = "10.1112/blms/bdr116",

language = "الإنجليزيّة",

volume = "44",

pages = "520--532",

journal = "Bulletin of the London Mathematical Society",

issn = "0024-6093",

publisher = "John Wiley and Sons Ltd",

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T1 - Hilbert series of PI relatively free G-graded algebras are rational functions

AU - Aljadeff, Eli

AU - Kanel-Belov, Alexei

N1 - 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).

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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U2 - 10.1112/blms/bdr116

DO - 10.1112/blms/bdr116

M3 - مقالة

SN - 0024-6093

VL - 44

SP - 520

EP - 532

JO - Bulletin of the London Mathematical Society

JF - Bulletin of the London Mathematical Society

IS - 3

ER -

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

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