ON SOME STANDARD GRADED ALGEBRAS IN MODULAR INVARIANT THEORY

Abstract

For a finite-dimensional representation V of a finite group G over a field K we denote the graded algebra R ≔ ⨁d≥0 Rd; where Rd ≔ ( Sym d∣G∣V*)G. We study the standardness of R for the representations [Formula: see text], [Formula: see text], and [Formula: see text], where Vn denote the n-dimensional indecomposable representation of the cyclic group Cp over the Galois field 𝔽p, for a prime p. We also prove the standardness for the defining representation of all finite linear groups with polynomial rings of invariants. This is motivated by a question of projective normality raised in [S. S. Kannan, S. K. Pattanayak and P. Sardar, Projective normality of finite groups quotients, Proc. Amer. Math. Soc.137(3) (2009) 863–867].