Upper triangular matrices and operations in odd primary connective K-theory

Abstract

Abstract We prove analogues for odd primes of results of Snaith and Barker–Snaith. Let ℓ denote the p-complete connective Adams summand and consider the group of left ℓ-module automorphisms of ℓ∧ℓ in the stable homotopy category which induce the identity on mod p homology. We prove a group isomorphism between this group and a certain group of infinite invertible upper triangular matrices with entries in the p-adic integers. We determine information about the matrix corresponding to the automorphism 1∧Ψ q of ℓ∧ℓ, where Ψ q is the Adams operation and q is an integer which generates the p-adic units.