Uniform Computability of Isotypic Block Diagonalization for Finite-Group Representations

Abstract

We study the universal isotypic block-diagonalization problem in the Type-2 theory of effectivity (TTE). We obtain a uniform computability result for the universal block diagonalizer for commutants of given finite group representations. As an application, when the given representation is orthogonal, we compute ordinary $QR$ factorizations blockwise in the isotypic basis. After transforming back to the original coordinates this yields a factorization $A=QX$, where $Q$ is orthogonal and $X$ is generally not upper triangular. Moreover, given a Wedderburn-adapted orthogonal basis, the transformed-back factors can be computed within the commutant.