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

TL;DR

This paper establishes a correspondence between automorphisms of p-complete connective Adams summand spectra and infinite upper triangular matrices over p-adic integers, extending known results to odd primes.

Contribution

It provides an analogue for odd primes of previous results relating automorphisms in stable homotopy to matrix groups, specifically for the p-complete connective Adams summand.

Findings

Group of automorphisms is isomorphic to a group of infinite upper triangular matrices.

Identifies the matrix corresponding to the automorphism 1 smash Psi^q.

Provides structural insights into automorphisms in odd primary connective K-theory.

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 smash 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 smash Psi^q, where Psi^q is the Adams operation and q is an integer which generates the p-adic units.