The Matrix Corresponding to 1 ∧ ψ³

Abstract

The objective of this chapter is to determine the conjugacy class of the map 1 ∧ψ³ in the upper triangular group U∞ℤ₂ in the sense of Chapter 3, Theorem 3.1.2. §1 recapitulates the background and states the main result (Theorem 5.1.2). §2 contains the central calculations in which the effect of 1 ∧ ψ³ is estimated with respect to the ℤ₂-module basis coming from the version of the Mahowald splitting given in Chapter 3. § 3 uses these calculations to determine the diagonal and super-diagonal elements in the matrix. § 4 gives two proofs of the result (Theorem 5.4.2) that any two matrices with this diagonal and super-diagonal are conjugate — in particular this gives Theorem 5.1.2. §5 contains two applications. The first describes the analogous upper triangular result in which bu ∧ bo is replaced by bu ∧ bu and the second uses some elementary matrix algebra to prove a result concerning the map φ_n : bo → bo given by φ_n = (ψ³-1) (ψ³-9) ... (ψ³-9^n-1), which is similar to a result of [185] concerning the Adams filtration of φ_n.