A classification of polynomial algebras as modules over the Steenrod algebra

Abstract

Suppose that A2 is the mod-2 Steenrod algebra, and that R = F2[x1, . . . , xn] is the polynomial ring in n variables over the prime field F2. Campbell and Selick [3] observed that the equations Sqxi = x 2 i−1 for 2≤ i≤ n, and Sqx1 = x 2 n, define an action of A2 on R that makes R isomorphic as an A2-module to the A2-module defined by the standard action on R. In that paper, they also pose the following question, due to Tom Hunter: does the equation Sqxi = ∑ jmijx 2 j (where M = [mij ] is any n-by-n matrix over F2) define an action of A2 on R? In this paper we give an affirmative answer to the above question and classify the actions defined in this way. More precisely, let S(V ) be the symmetric