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Product of two involutions in quaternionic special linear group

Part of: Basic linear algebra Special matrices Structure and classification of infinite or finite groups Other groups of matrices

Published online by Cambridge University Press:  08 January 2025

Krishnendu Gongopadhyay Open the ORCID record for Krishnendu Gongopadhyay [Opens in a new window] ,
Tejbir Lohan Open the ORCID record for Tejbir Lohan [Opens in a new window]  and
Chandan Maity Open the ORCID record for Chandan Maity [Opens in a new window]
Krishnendu Gongopadhyay
Affiliation:
Indian Institute of Science Education and Research (IISER) Mohali, Knowledge City, Sector 81, S.A.S. Nagar 140306, Punjab, India e-mail: krishnendu@iisermohali.ac.in, krishnendug@gmail.com
Tejbir Lohan
Affiliation:
Indian Institute of Technology Kanpur, Kanpur 208016, Uttar Pradesh, India e-mail: tejbirlohan70@gmail.com tejbir@iitk.ac.in
Chandan Maity*
Affiliation:
Indian Institute of Science Education and Research (IISER) Berhampur, Berhampur 760003, Odisha, India
*
e-mail: cmaity@iiserbpr.ac.in
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Abstract

An element of a group is called reversible if it is conjugate to its own inverse. Reversible elements are closely related to strongly reversible elements, which can be expressed as a product of two involutions. In this paper, we classify the reversible and strongly reversible elements in the quaternionic special linear group $ \mathrm {SL}(n,\mathbb {H})$ and quaternionic projective linear group $ \mathrm {PSL}(n,\mathbb {H})$. We prove that an element of $ \mathrm {SL}(n,\mathbb {H})$ (resp. $ \mathrm {PSL}(n,\mathbb {H})$) is reversible if and only if it is a product of two skew-involutions (resp. involutions).

Keywords

Reversible elementsstrongly reversible elementsquaternionic special linear groupWeyr canonical formreversing symmetry group

MSC classification

Primary: 20E45: Conjugacy classes 15B33: Matrices over special rings (quaternions, finite fields, etc.)
Secondary: 15A21: Canonical forms, reductions, classification 20H25: Other matrix groups over rings

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 68 , Issue 2 , June 2025 , pp. 421 - 439
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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