Steenrod operations on the modular invariants

Abstract

In this paper, we compute the action of the mod p Steenrod operations on the modular invariants of the linear groups with p an odd prime number. Introduction Fix an odd prime p. Let Apn be the alternating group on p letters. Denote by Σpn,p a Sylow p-subgroup of Apn and E an elementary abelian p-group of rank n. Then we have the restriction homomorphisms Res(E, Σpn,p) : H (BΣpn,p) −→ H (BE), Res(E, Apn) : H (BApn ) −→ H (BE), induced by the regular permutation representation E ⊂ Σpn,p ⊂ Apn of E (see Mùi [4]). Here and throughout the paper, we assume that the coefficients are taken in the prime field Z/p. Using modular invariant theory of linear groups, Mùi proved in [3, 4] that ImRes(E, Σpn,p) = E(U1, . . . , Un) ⊗ P (V1, . . . , Vn), ImRes(E, Apn ) = E(M̃n,0, . . . , M̃n,n−1) ⊗ P (L̃n, Qn,1, . . . , Qn,n−1), Here and in what follows, E(., . . . , .) and P (., . . . , .) are the exterior and polynomial algebras over Z/p generated by the variables indicated. L̃n, Q,s are the Dickson invariants of dimensions p, 2(p − p), and M̃n,s, , Uk, Vk are the Mùi invariants of dimensions p − 2p, pk−1, 2pk−1 respectively (see Section 1). Let A be the mod p Steenrod algebra and let τs, ξi be the Milnor elements of dimensions 2p − 1, 2p − 2 respectively in the dual algebra A∗ of A. In [7], Milnor showed that, as an algebra A∗ = E(τ0, τ1, . . .) ⊗ P (ξ1, ξ2, . . .). Then A∗ has a basis consisting of all monomials τSξ = τs0 . . . τsk ξ r1 . . . ξm , with S = (s1, . . . , sk), 0 6 s1 < . . . < sk, R = (r1, . . . , rm), ri > 0. Let St ∈ A denote the dual of τSξ with respect to that basis. Then A has a basis consisting all operations St. For S = ∅, R = (r), St∅,(r) is nothing but the Steenrod operation P . Since H(BG), G = E, Σpn,p or Apn , is an A-module (see [13, Chap. VI]) and the restriction homomorphisms are A-linear, their images are A-submodules of H(BE). 2010 Mathematics Subject Classification. Primary 55S10; Secondary 55S05.