TY - JOUR

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.

UR - http://www.scopus.com/inward/record.url?scp=84861580984&partnerID=8YFLogxK

U2 - 10.1112/blms/bdr116

DO - 10.1112/blms/bdr116

M3 - 文章

AN - SCOPUS:84861580984

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 -