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Conjugate real classes in general linear groups

Krishnendu Gongopadhyay

Indian Institute of Science Education and Research (IISER) Mohali, Sector 81, SAS Nagar, Punjab 140306, India

E-mail Address: krishnendu@iisermohali.ac.in

E-mail Address: krishnendug@gmail.com

Corresponding author.

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Sudip Mazumder

Department of Mathematics, Jadavpur University, Jadavpur 700032, Kolkata, India

E-mail Address: topologysudip@gmail.com

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Sujit Kumar Sardar

Department of Mathematics, Jadavpur University, Jadavpur 700032, Kolkata, India

E-mail Address: sksardarjumath@gmail.com

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https://doi.org/10.1142/S0219498819500543Cited by:2 (Source: Crossref)

Abstract

Let $F$ $F$ be a field with a nontrivial involution $c : α \mapsto α^{c}$ $c : α \mapsto α^{c}$ . An element $g \in {GL}_{n} (F)$ $g \in {GL}_{n} (F)$ is called $c$ $c$ -real if it is conjugate to ${(g^{c})}^{- 1}$ ${(g^{c})}^{- 1}$ . We prove that for $n \geq 2$ $n \geq 2$ , $g \in {GL}_{n} (F)$ $g \in {GL}_{n} (F)$ is $c$ $c$ -real if and only if it has a representation in some unitary group of degree $n$ $n$ over $F$ $F$ .

Communicated by E. Vdovin

Keywords:

AMSC: 20E45, 20G15, 15A04

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