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Strongly reversible classes in SL(n, ℂ)

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

Published online by Cambridge University Press:  30 April 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:
Department of Mathematical Sciences, Indian Institute of Science Education and Research (IISER) Mohali, Sector 81, S.A.S. Nagar, Knowledge City, Punjab 140306, India
Tejbir Lohan
Affiliation:
Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, Uttar Pradesh 208016, India
Chandan Maity
Affiliation:
Department of Mathematical Sciences, Indian Institute of Science Education and Research (IISER) Berhampur, Berhampur, Odisha 760003, India
*
Corresponding author: Krishnendu Gongopadhyay, email: krishnendu@iisermohali.ac.in
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Abstract

An element of a group is called strongly reversible or 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 $\mathrm{SL}(n,\mathbb{C})$.

Keywords

reversibilitystrongly reversible elementsbireflectional elementsreversing symmetry groupJordan canonical formWeyr canonical form

MSC classification

Primary: 20E45: Conjugacy classes 15A23: Factorization of matrices
Secondary: 15A21: Canonical forms, reductions, classification 15A27: Commutativity

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 68 , Issue 4 , November 2025 , pp. 1137 - 1168
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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References

Baake, M. and Roberts, J. A. G., The structure of reversing symmetry groups, Bull. Austral. Math. Soc. 73(3): (2006), 445–459.CrossRefGoogle Scholar
Basmajian, A. and Maskit, B., Space form isometries as commutators and products of involutions, Trans. Amer. Math. Soc. 364(9): (2012), 5015–5033.CrossRefGoogle Scholar
Burton, D. M., Elementary Number theory, 7th ed. (McGraw-Hill, New York, 2011).Google Scholar
Djoković, D. Ž., Product of two involutions, Arch. Math. (Basel) 18 (1967), 582–584.CrossRefGoogle Scholar
Gill, N. and Singh, A., Real and strongly real classes in $\mathrm{SL}_{n}(q)$, J. Group Theory 14(3): (2011), 437–459.Google Scholar
Gohberg, I., Lancaster, P. and Rodman, L., Invariant Subspaces of Matrices with Applications. Republication SIAM (John Wiley, Philadelphia, 2006).Google Scholar
Gongopadhyay, K., Lohan, T. and Maity, C., Reversibility of affine transformations, Proc. Edinb. Math. Soc (2) 66(4): (2023), 1217–1228.CrossRefGoogle Scholar
Gongopadhyay, K. and Maity, C., Reality of unipotent elements in classical Lie groups, Bull. Sci. Math. 185(103261): (2023), 29.CrossRefGoogle Scholar
Gow, R., Products of two involutions in classical groups of characteristic 2, J. Algebra 71(2): (1981), 583–591.CrossRefGoogle Scholar
Gustafson, W. H., Halmos, P. R. and Radjavi, H., Products of involutions, Linear Algebra Appl. 13(1–2): (1976), 157–162.CrossRefGoogle Scholar
Hoffman, F. and Paige, E. C., Products of two involutions in the general linear group, Indiana Univ. Math. J. 20 (11): (1970/71), 1017–1020.CrossRefGoogle Scholar
Knüppel, F. and Nielsen, K., SL(V) is 4-reflectional, Geom. Dedicata 38(3): (1991), 301–308.CrossRefGoogle Scholar
Liu, K. M., Decomposition of matrices into three involutions, Linear Algebra Appl. 111 (1988), 1–24.CrossRefGoogle Scholar
O’Farrell, A. G. and Short, I., Reversibility in Dynamics and Group Theory. London Mathematical Society Lecture Note Series, Volume 416 (Cambridge University Press, Cambridge, 2015).Google Scholar
O’Meara, K. C., Clark, J. and Vinsonhaler, C. I., Advanced Topics in Linear Algebra: Weaving Matrix Problems Through the Weyr Form (Oxford University Press, Oxford, 2011).Google Scholar
Paras, A. T. and Salinasan, J. R., The product of an involution and a skew-involution, Linear Algebra Appl. 584 (2020), 431–437.CrossRefGoogle Scholar
Pazzis, C. S., Products of involutions in the stable general linear group, J. Algebra 530 (2019), 235–289.CrossRefGoogle Scholar
Pazzis, C. S., Products of involutions of an infinite-dimensional vector space, Canad. J. Math. 73(1): (2021), 195–220.CrossRefGoogle Scholar
Roche, A. and Vinroot, C. R., A factorization result for classical and similitude groups, Canad. Math. Bull. 61(1): (2018), 174–190.CrossRefGoogle Scholar
Shapiro, H., The Weyr characteristic, Am. Math. Mon. 106(10): (1999), 919–929.CrossRefGoogle Scholar
Shapiro, H., Linear Algebra and Matrices: Topics for a Second Course. Pure and Applied Undergraduate Texts, 24 (American Mathematical Society, Providence, RI, 2015).Google Scholar
Singh, A. and Thakur, M., Reality properties of conjugacy classes in algebraic groups, Israel J. Math. 165 (2008), 1–27.CrossRefGoogle Scholar
Wonenburger, M. J., Transformations which are products of two involutions, J. Math. Mech 16(4): (1966), 327–338.Google Scholar