The finite irreducible linear groups with polynomial ring of invariants (Q1383782)

Let \(V\) be a finite dimensional vector space over a field \(K\) and let \(S(V)\) denote the symmetric algebra of \(V\). If \(n=\dim V\) then \(S(V)\) is isomorphic to the polynomial algebra of \(n\) variables over \(K\). Let \(G\subset \text{GL} (V)\) be a finite group. Then the action of \(G\) on \(V\) extends to \(S(V)\). Set \(S(V)^G= \{s\in S(V): g(s) =s\) for all \(g\in G\}\). The problem of determining \(G\) such that \(S(V)^G\) is isomorphic to a polynomial algebra has a long history. If \(\text{char} (K)=0\), the problem was solved in the 50th. This paper provides a full solution of the problem for the case where \(\text{char} (K)= p>0\). A necessary condition was known much earlier: If \(S(V)^G\) is a polynomial algebra then so is \(S(V)^{C(W)}\) for each subspace \(W\subset V\) (here \(C(W)\) denote the centralizer of \(W\) in \(G)\). The main result of the paper under review proves that this condition is sufficient. If \(G\) contains no transvection, then \(S(V)^G\) is a polynomial algebra if \(C(W)\) is generated by pseudoreflections.