On R. Steinberg's theorem on algebras of coinvariants
-
Larry Smith
Abstract
Steinberg's theorem on the coinvariant algebra 
of a complex representation 
of a finite group G says that is a Poincaré duality algebra if and only if the invariant algebra 
is a polynomial algebra. The extension of this to the nonmodular case has been achieved in stages, the final result being obtained by W.G. Dwyer and C.W. Wilkerson. We show that the main module theoretic tool they use extends to the following characteristic free result: If 𝔽[V]G is a Poincaré duality algebra of formal dimension d, then 𝔽[V]G is a polynomial algebra if and only if 
contains a nonzero element of degree –d. In the nonmodular case an easy transfer argument then recovers their extension of Steinberg's theorem by means of some representation theory. Combined with some new results concerning the Δ operators of Demazure, our characteristic free result yields the following for reflection groups: A reflection group G for which 𝔽[V]G is a Poincaré duality algebra in which the trivial G-representation 1G occurs only once as a subrepresentation has a polynomial algebra for its invariant algebra 𝔽[V]G.
© de Gruyter 2009
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- A Hopf theorem for open surfaces in product spaces
- On R. Steinberg's theorem on algebras of coinvariants
- Natural pseudo-distances between closed curves
- Clifford semigroups of ideals in monoids and domains
- Lp norm estimates of eigenfunctions restricted to submanifolds
- Central values of generalized multiple sine functions
- Lp-independence of spectral bounds of non-local Feynman-Kac semigroups
- On pairwise mutually permutable products
- The block structure spaces of real projective spaces and orthogonal calculus of functors II
- Erratum to: “Intrinsic ultracontractivity for non-symmetric Lévy processes” [Forum Math. 21 (2009) 43–66]
Articles in the same Issue
- A Hopf theorem for open surfaces in product spaces
- On R. Steinberg's theorem on algebras of coinvariants
- Natural pseudo-distances between closed curves
- Clifford semigroups of ideals in monoids and domains
- Lp norm estimates of eigenfunctions restricted to submanifolds
- Central values of generalized multiple sine functions
- Lp-independence of spectral bounds of non-local Feynman-Kac semigroups
- On pairwise mutually permutable products
- The block structure spaces of real projective spaces and orthogonal calculus of functors II
- Erratum to: “Intrinsic ultracontractivity for non-symmetric Lévy processes” [Forum Math. 21 (2009) 43–66]
