Poincaré duality algebras and rings of coinvariants

Let G be a finite group acting on the n-dimensional vector space V = Fn over the field F via a faithful representation :G ! GL(n, F). Then G also acts on the algebra F[V ] of homogeneous polynomial functions on V . R. Steinberg [Trans. Amer. Math. Soc. 112 (1964), 392–400; MR0167535 (29 #4807)] proved that if F has characteristic 0, then the ring of coinvariants F[V ]G is a Poincar´e duality algebra if an only if G is a pseudoreflection group. The main result of the present paper is that an analogue of Steinberg’s theorem holds when F has positive characteristic p and the order of G is relatively prime to p (see also [L. Smith, Proc. Amer. Math. Soc. 131 (2003), no. 4, 1043–1048 (electronic); MR1948093 (2003k:13008)] for Steinberg’s theorem in positive characteristic). To prove this result the author uses a theorem by L. Smith [Quart. J. Math. Oxford Ser. (2) 33 (1982), no. 131, 379–384; MR0668184 (84a:57044a)] and a process based on the Witt ringW(F) of a perfect field F.