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The combinatorics of permuting and preserving curve-bound spectra

Alexandru Chirvasitu

TL;DR

This paper classifies continuous maps that preserve both spectrum and commutativity from curve-constrained matrix spaces ${\mathcal{H}}_{n|\Lambda}$ to ${\mathcal M}_n({\mathbb C})$, where $\Lambda$ is a simple curve and $n\ge 3$. It extends results of Petek and prior work on Hermitian matrices by showing such maps are restricted to a small set of canonical forms: conjugation, transpose-conjugation, or a spectrum-ordered embedding aligned with the orientation of $\Lambda$ (with fixed eigenspaces). The authors develop a combinatorial, Grassmannian-based approach using spectral projections ${\mathcal K}_{S}(T)$ and tableau decompositions ${\mathcal K}_{\overset{\circ}{\mu}}(T)$ to construct a reconstruction map $\Phi$ on Grassmannians, which anchors the classification to the action on eigen-subspaces. When $\Lambda$ is a simple closed curve, the spectrum-ordering option is ruled out, so only conjugation and transpose-conjugation remain, while continuity and regularity of $\Lambda$ govern the emergence of additional involution-type possibilities in broader settings.

Abstract

We prove that continuous spectrum- and commutativity-preserving maps to $\mathcal{M}_n(\mathbb{C})$ from the space of normal (real or complex) $n\times n$, $n\ge 3$ matrices with spectra contained in a given continuous-injection interval image $Λ\subseteq \mathbb{C}$ or $\mathbb{R}$ are (a) conjugations; (b) transpose conjugations, or (c) orderings of spectra according to an orientation of $Λ$, with fixed eigenspaces. This generalizes results of Petek's (self-maps of real or complex Hermitian matrices) and the author's (complex Hermitian matrices as the domain, $\mathcal{M}_n(\mathbb{C})$ as the codomain). An application rules out possibility (c) for normal matrices with spectra constrained to a simple closed curve, extending a result by the author, Gogić and Tomašević to the effect that continuous commutativity and spectrum preservers on unitary groups are (transpose) conjugations. The involution preserving eigenspaces and complex-conjugating eigenvalues is a novel possibility beyond (a), (b) and (c) if the domain consists of all semisimple operators with $Λ$-bound spectra instead; its continuity (or lack thereof) and whether or not that map furthermore extends continuously to arbitrary $Λ$-constrained-spectrum matrices hinge on the geometry and regularity of $Λ$.

The combinatorics of permuting and preserving curve-bound spectra

TL;DR

This paper classifies continuous maps that preserve both spectrum and commutativity from curve-constrained matrix spaces to , where is a simple curve and . It extends results of Petek and prior work on Hermitian matrices by showing such maps are restricted to a small set of canonical forms: conjugation, transpose-conjugation, or a spectrum-ordered embedding aligned with the orientation of (with fixed eigenspaces). The authors develop a combinatorial, Grassmannian-based approach using spectral projections and tableau decompositions to construct a reconstruction map on Grassmannians, which anchors the classification to the action on eigen-subspaces. When is a simple closed curve, the spectrum-ordering option is ruled out, so only conjugation and transpose-conjugation remain, while continuity and regularity of govern the emergence of additional involution-type possibilities in broader settings.

Abstract

We prove that continuous spectrum- and commutativity-preserving maps to from the space of normal (real or complex) , matrices with spectra contained in a given continuous-injection interval image or are (a) conjugations; (b) transpose conjugations, or (c) orderings of spectra according to an orientation of , with fixed eigenspaces. This generalizes results of Petek's (self-maps of real or complex Hermitian matrices) and the author's (complex Hermitian matrices as the domain, as the codomain). An application rules out possibility (c) for normal matrices with spectra constrained to a simple closed curve, extending a result by the author, Gogić and Tomašević to the effect that continuous commutativity and spectrum preservers on unitary groups are (transpose) conjugations. The involution preserving eigenspaces and complex-conjugating eigenvalues is a novel possibility beyond (a), (b) and (c) if the domain consists of all semisimple operators with -bound spectra instead; its continuity (or lack thereof) and whether or not that map furthermore extends continuously to arbitrary -constrained-spectrum matrices hinge on the geometry and regularity of .
Paper Structure (1 section, 9 theorems, 16 equations)

This paper contains 1 section, 9 theorems, 16 equations.

Table of Contents

  1. Curve-constrained generalized Hermitian matrices

Key Result

Theorem 2

For $n\in {\mathbb Z}_{\ge 3}$ and a simple curve $I\lhook\joinrel\xrightarrow{\iota}\Bbbk\in \left\{{\mathbb R},{\mathbb C}\right\}$ the continuous, commutativity- and spectrum-preserving maps ${\mathcal{H}}^*_{n\mid \gamma}(\Bbbk)\xrightarrow{\phi}{\mathcal{M}}_n({\mathbb C})$ are precisely those

Theorems & Definitions (15)

  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Remark 1.1
  • Definition 1.2
  • Proposition 1.6
  • Proof 1
  • Proposition 1.7
  • Proof 2
  • Corollary 1.8
  • ...and 5 more