\(\psi^{3}\) as an upper triangular matrix (Q864961)

The Adams operation \(\psi^3\) generates the group of self-equivalences of the ring spectrum \(bo\) associated to \(2\)-complete connective real \(K\)-theory. The main result of this paper is to calculate the effect of this map in \(bu_\ast bo\), where \(bu_\ast\) is connective complex \(K\)-theory. More explicitly, the authors give a description of the map \(1 \wedge \psi^3\) on the spectrum \(bu \wedge bo\). This is a self-equivalence of \(bu \wedge bo\) as a left \(bU\)-module. By results of Mahowald, such an equivalence can be expressed as an invertible, infinite, upper triangular matrix with coefficients in the \(2\)-adic integers, and the authors compute this matrix. The answer is quite pleasant, with increasing powers of \(9\) down the diagonal and \(1\)s just above it. All other entries are zero. Some sample applications are given in the last section. The paper is explicitly computational; however, the section headings provide a running joke for those familiar with the \textit{Matrix} trilogy of movies.