A primer on the Dickson invariants

Abstract

The ring of invariant forms of the full linear group GL(V ) of a finite dimensional vector space over the finite field Fq was computed early in the 20th century by L. E. Dickson [5], and was found to be a graded polynomial algebra on certain generators {cn,i}. This ring of invariants, for q = p, has found use in algebraic topology in work of Milgram–Man [9], Singer [16,17], Adams–Wilkerson [1], Rector [13], Lam [6], Mui [11], and Smith–Switzer [18]. The aim of this exposition is to give a simple proof of the structure of the ring of invariants, and to compute the action of the Steenrod algebra on the generators of the invariants. The methods used are implicit in Adams–Wilkerson. Dickson’s viewpoint was to vastly generalize the defining equation of Fq,