Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Invariant embeddings and weighted permutations

Mitja Mastnak, Heydar Radjavi

Abstract

We prove that for any fixed unitary matrix $U$, any abelian self-adjoint algebra of matrices that is invariant under conjugation by $U$ can be embedded into a maximal abelian self-adjoint algebra that is still invariant under conjugation by $U$. We use this result to analyse the structure of matrices $A$ for which $A^*A$ commutes with $AA^*$, and to characterize matrices that are unitarily equivalent to weighted permutations.

Invariant embeddings and weighted permutations

Abstract

We prove that for any fixed unitary matrix , any abelian self-adjoint algebra of matrices that is invariant under conjugation by can be embedded into a maximal abelian self-adjoint algebra that is still invariant under conjugation by . We use this result to analyse the structure of matrices for which commutes with , and to characterize matrices that are unitarily equivalent to weighted permutations.
Paper Structure (4 sections, 9 theorems, 8 equations)

This paper contains 4 sections, 9 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Invariant embeddings of commutative semi-simple algebras
  4. Half-normal operators and weighted permutations

Key Result

Theorem 1

Let $\mathcal{G}$ be a group. If every nonabelian sub-quotient of $\mathcal{G}$ contains a noncentral abelian normal subgroup, then every irreducible reprentation of every subgroup is monomializable.

Theorems & Definitions (18)

  • Theorem 1: Brall
  • proof
  • Theorem 2
  • proof
  • Corollary 3
  • Corollary 4
  • Corollary 5
  • Remark 6
  • Lemma 7
  • proof
  • ...and 8 more