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Strongly reversible classes in $\mathrm{SL}(n,\mathbb{C})$

Krishnendu Gongopadhyay, Tejbir Lohan, Chandan Maity

TL;DR

The work delivers a complete classification of strongly reversible elements in ${$\mathrm{SL}}(n,\mathbb{C})$ by combining Jordan and Weyr form analyses with reversing symmetry groups. It shows that an element $A$ reversible in ${$\mathrm{SL}}(n,\mathbb{C})$ is strongly reversible iff either a block $\mathrm{J}(\mu,2r+1)$ with $\mu\in\{-1,+1\}$ exists or the parity condition on block counts $s+t$ is even, where $s$ counts blocks ${\mathrm{J}}(\mu,4k+2)$ and $t$ counts pairs ${\mathrm{J}}(\lambda,2m+1),{\mathrm{J}}(\lambda^{-1},2m+1)$. The paper also treats semisimple and unipotent cases, provides explicit reversers via matrices like $\Omega(\lambda,n)$ and Weyr-based constructions, and highlights the role of centralizers in GL and SL contexts. The results extend prior partial classifications and offer a concrete framework for determining strong reversibility in algebraically closed fields of characteristic not 2, with potential extensions to related affine groups. Overall, the paper delivers a precise, constructive criterion for strong reversibility in ${$\mathrm{SL}}(n,\mathbb{C})$ and advances understanding of reversing symmetry structures in linear groups.

Abstract

An element of a group is called $\textit{strongly reversible}$ or $\textit{strongly real}$ if it can be expressed as a product of two involutions. We provide necessary and sufficient conditions for an element of $\mathrm{SL}(n,\mathbb{C})$ to be a product of two involutions. In particular, we classify the strongly reversible conjugacy classes in $\mathrm{SL}(n,\mathbb{C})$.

Strongly reversible classes in $\mathrm{SL}(n,\mathbb{C})$

TL;DR

The work delivers a complete classification of strongly reversible elements in \mathrm{SL}}(n,\mathbb{C})A{ is strongly reversible iff either a block with exists or the parity condition on block counts is even, where counts blocks and counts pairs . The paper also treats semisimple and unipotent cases, provides explicit reversers via matrices like and Weyr-based constructions, and highlights the role of centralizers in GL and SL contexts. The results extend prior partial classifications and offer a concrete framework for determining strong reversibility in algebraically closed fields of characteristic not 2, with potential extensions to related affine groups. Overall, the paper delivers a precise, constructive criterion for strong reversibility in \mathrm{SL}}(n,\mathbb{C})$ and advances understanding of reversing symmetry structures in linear groups.

Abstract

An element of a group is called or if it can be expressed as a product of two involutions. We provide necessary and sufficient conditions for an element of to be a product of two involutions. In particular, we classify the strongly reversible conjugacy classes in .
Paper Structure (14 sections, 23 theorems, 93 equations, 1 table)

This paper contains 14 sections, 23 theorems, 93 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation for partition of $n$
  4. Block Matrices
  5. Jordan canonical form
  6. Weyr canonical form
  7. Duality between the Jordan and Weyr Forms
  8. Reversing symmetry groups in ${\rm GL}(n,\mathbb {C})$
  9. Reverser set of certain Jordan forms
  10. Strong reversibility of certain Jordan forms in ${\rm SL}(n,\mathbb {C})$
  11. Strongly reversible semisimple elements
  12. Strong reversibility of unipotent elements in ${\rm SL}(n,\mathbb {C})$
  13. Strong reversibility in ${\rm SL}(n,\mathbb {C})$
  14. Proof of Theorem \ref{['thr-main-strong-rev-SL(n, C)']}

Key Result

Theorem 1.1

An element $A \in {\rm SL }(n,\mathbb {C})$ is reversible if and only if the Jordan blocks in the Jordan decomposition of $A$ can be partitioned into singletons $\{\mathrm{J}(\mu, k )\}$ or pairs $\{ \mathrm{J}(\lambda, m),\mathrm{J}(\lambda^{-1}, m)\}$, where $\mu, \lambda \in \mathbb {C} \setminu

Theorems & Definitions (41)

  • Theorem 1.1: OS
  • Theorem 1.2
  • Definition 1.3
  • Definition 2.1: cf. COV
  • Definition 2.2
  • Lemma 2.3
  • Definition 2.4: GLR
  • Lemma 2.5: Jordan form in $\mathrm{M}(n,\mathbb {C})$, GLR
  • Definition 2.6: COV
  • Definition 2.7: COV
  • ...and 31 more