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 - https://doi.org/10.1112/blms/bdr116
DO - https://doi.org/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 -