Vector Invariants ofU2(Fp) : A Proof of a Conjecture of Richman

AbstractIn this paper we prove a conjecture of D. R. Richman concerning the vector invariants of the groupU2(Fp). LetVbe a vector space of dimension 2 with basisx,yover the fieldFpand letFp[x, y] be the symmetric algebra ofVoverFp. Ifσdenotes a generator ofU2(Fp) then we may assumeσ(x)=xandσ(y)=x+y. LetAnbe the symmetric algebra ofV⊕n. We obtain an automorphism ofAnof order p by using the diagonal action ofσextended to the whole ofAn. The subalgebra of polynomials left invariant by this action is called the ring of vector invariants ofU2(Fp). Richman conjectured that these rings of invariants have certain sets of generators and gave a proof in the casep=2. We prove his conjecture for all primes