Upper triangular matrices and operations in odd primary connective \(K\)-theory (Q2343347)

By analogues for odd primes of the results of Snaith and Barker-Snaith, let \(\ell\) denote the Adams summand. The exterior square of \(\ell\) can be decomposed as \[ \ell \wedge \ell = \ell \wedge \bigvee_{n\geq 0} \mathcal K(n), \] where the spectra \(\mathcal K(n)\) are suspensions of Brown-Gitler spectra, realising a weight filtration of the homology of \(\Omega^2S^3\langle 3\rangle_p\) providing \(p\)-complete finite spectra. Denote by \(\text{Aut}^{0}_{\text{left}- \ell-\text{mod}}(\ell\wedge \ell)\) the group of automorphisms of left-\(\ell\)-modules (i.e. homotopy equivalences) which induce the identity map in mod \(p\) homology. The main result of the paper is Theorem 2.4 stating that this group \(\text{Aut}^0_{\text{left}-\ell-\text{mod}}(\ell\wedge \ell)\) is isomorphic to the group \(U_\infty\mathbb Z_p\) of upper triangular \(p\)-adic integral matrices the diagonal entries of which are lying in the subgroup \(1+p\mathbb Z_p\). The isomorphism can be modified so that for the Adams operation \(\Psi^q\), the left-\(\ell\)-module automorphism \(1\wedge \Psi^q\) is of specific infinite matrix form \(R\) with the only non-zero entries \(1, \hat{q}, \hat{q}^2, \dots\) on the diagonal and with 1's on the one-line upper diagonal places (Theorem 4.6), where \(\hat{q} = q^{p-1}\).