Vector invariants of U-2(F-p): A proof of a conjecture of Richman

In this paper we prove a conjecture of D. R. Richman concerning the vector invariants of the group U-2(F-p). Let V be a vector space of dimension 2 with basis x, y over the field F-p and let F-p[x, y] be the symmetric algebra of V over F-p. If sigma denotes a generator of U-2(F-p) then we may assume sigma(x) = x and sigma(y) = x + y. Let A(n) be the symmetric algebra of V+n. We obtain an automorphism of A(n) of order p by using the diagonal action of sigma extended to the whole of A(n). The subalgebra of polynomials left invariant by this action is called the ring of vector invariants of U-2(F-2). Richman conjectured that these rings of invariants have certain sets of generators and gave a proof in the case p=2. We prove his conjecture for all primes. (C) 1997 Academic Press.