Upper Triangular Matrices and Operations in Odd Primary Connective K-Theory

Abstract

Let $U_\infty\Z_p$ be the group of infinite invertible upper triangular matrices with entries in the $p$-adic integers. Also let $\Aut_{\text{left-}\ell\text{-mod}}^0(\ell\wedge\ell)$ be the group of left $\ell$-module automorphisms of $\ell\wedge\ell$ which induce the identity on mod $p$ homology, where $\ell$ is the Adams summand of the $p$-adically complete connective $K$-Theory spectrum. In this thesis we construct and prove there is an isomorphism between these two groups. We will then determine a specific matrix (up to conjugacy) which corresponds to the automorphism $1\wedge\psi^q$ of $\ell\wedge\ell$ where $\psi^q$ is the Adams operation and $q$ is an integer which generates the $p$-adic units $\Z_p^\times$.

We go on to look at the map $1\wedge\phi_n$ where $\phi_n=(\psi^q-1)(\psi^q-r)\cdots(\psi^q-r^{n-1})$ and $r=q^{p-1}$ under a generalisation of the map which gave us the isomorphism. Lastly we use some of the ideas presented to give us a new way of looking at the ring of degree zero operations on the connective $p$-local Adams summand via upper triangular matrices.