On the ring of invariants of \(F^*_{2^ n}\)
Let \(\{x_{n-1},...,x_ 0\}\) be an \({\mathbb{F}}_ 2\)-basis for V and \(W\equiv {\mathbb{F}}_{2^ n}\otimes_{{\mathbb{F}}_ 2}V\). The cyclic group G generated by the diagonal matrix \((\omega^{2^{n-1}},\omega^{2^{n- 2}},...,\omega)\) acts on W, where \(\omega\) is a generator of \({\mathbb{F}}^*_{2^ n}\), and setting \(P_ n={\mathbb{F}}_ 2[V]\) where each \(| x_ i| =1\), it also acts on \(Q_ n\equiv {\mathbb{F}}_{2^ n}\otimes_{{\mathbb{F}}_ 2}P_ n\). The paper gives a description of a minimal generating set for \(Q^ G_ n\) and for \(P^ G_ n\equiv P_ n\cap Q^ G_ n\), and formulae for the number of elements involved. Further, defining \(Sq^ 1x_ i=x^ 2_{i-1}\), \(Sq^ jx_ i=0\) for \(j>1\) and requiring \({\mathbb{F}}_{2^ n}\)-linearity, gives an action of the Steenrod algebra A on \(P_ n\) and \(Q_ n\) (where the former is isomorphic to the standard A-module structure and the latter action commutes with the action of G). The action of A extends to \(S_ n\) and \(R_ n\), the rings of Laurent polynomials associated with \(Q_ n\) and \(P_ n\). It is established that the unstable elements of \(S_ n\), \(R_ n\), \(S^ G_ n\) and \(R^ G_ n\equiv S^ G_ n\wedge R_ n\) are precisely \(Q_ n\), \(P_ n\), \(Q^ G_ n\) and \(P^ G_ n\), respectively.
scientific article; zbMATH DE number 4209092