arXiv subjects

Tejbir Lohan

Publications and source records attributed to Tejbir Lohan.

13 recordsLinked to original sources

arXiv2026

Selected facts on products of two involutions in the Riordan group

An element of a group is called \emph{reversible} if it is conjugate to its inverse, and \emph{strongly reversible} if it can be expressed as a product of two involutions. We study strongly reversible elements in the Riordan group and in several of its important subgroups. We show that not every reversible element in the Riordan group is strongly reversible, and we investigate products of reversible elements in the Riordan group.

math.GR↗

arXiv2026

Real characters and real classes of $\mathrm{GL}_2$ and $\mathrm{GU}_2$ over discrete valuation rings

Let $\mathfrak{o}$ be the ring of integers of a non-archimedean local field with residue field of odd characteristic, $\mathfrak{p}$ be its maximal ideal and let $\mathfrak{o}_\ell = \mathfrak{o}/\mathfrak{p}^\ell$ for $\ell\ge 2$. In this article, we study real-valued characters and real representations of the finite groups $\mathrm{GL}_2(\mathfrak{o}_\ell)$ and $\mathrm{GU}_2(\mathfrak{o}_\ell)$. We give a complete classification of real and strongly real classes of these groups and characterize the real-valued irreducible complex characters. We prove that every real-valued irreducible complex character of $\mathrm{GL}_2(\mathfrak{o}_\ell)$ is afforded by a representation over $\mathbb{R}$. In contrast, we show that $\mathrm{GU}_2(\mathfrak{o}_\ell)$ admits real-valued irreducible characters that are not realizable over $\mathbb{R}$. These results extend the parallel known phenomena for the finite groups $\mathrm{GL}_n(\mathbb{F}_q)$ and $\mathrm{GU}_n(\mathbb{F}_q)$.

math.RT↗

arXiv2025

Linear maps preserving product of involutions

An element of the algebra $M_n(\mathbb{F})$ of $n \times n$ matrices over a field $\mathbb{F}$ is called an involution if its square equals the identity matrix. Gustafson, Halmos, and Radjavi proved that any product of involutions in $M_n(\mathbb{F})$ can be expressed as a product of at most four involutions. In this article, we investigate the bijective linear preservers of the sets of products of two, three, or four involutions in $M_n(\mathbb{F})$.

math.FA↗

arXiv2024

Strongly real adjoint orbits of complex symplectic Lie group

We consider the adjoint action of the symplectic Lie group $\mathrm{Sp}(2n,\mathbb{C})$ on its Lie algebra $\mathfrak{sp}(2n,\mathbb{C})$. An element $X \in \mathfrak{sp}(2n,\mathbb{C})$ is called $\mathrm{Ad}_{\mathrm{Sp}(2n,\mathbb{C})}$-real if $ -X = \mathrm{Ad}(g)X$ for some $g \in \mathrm{Sp}(2n,\mathbb{C})$. Moreover, if $ -X = \mathrm{Ad}(h)X $ for some involution $h \in \mathrm{Sp}(2n,\mathbb{C})$, then $X \in \mathfrak{sp}(2n,\mathbb{C})$ is called strongly $\mathrm{Ad}_{\mathrm{Sp}(2n,\mathbb{C})}$-real. In this paper, we prove that for every element $X \in \mathfrak{sp}(2n,\mathbb{C})$, there exists a skew-involution $g \in \mathrm{Sp}(2n,\mathbb{C})$ such that $-X =\mathrm{Ad}(g)X$. Furthermore, we classify the strongly $\mathrm{Ad}_{\mathrm{Sp}(2n,\mathbb{C})}$-real elements in $\mathfrak{sp}(2n,\mathbb{C})$. We also classify skew-Hamiltonian matrices that are similar to their negatives via a symplectic involution.

math.GR↗

arXiv2024

Product of two involutions in quaternionic special linear group

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).

math.GR↗

arXiv2024

Algebraic characterization of reversibility in the quaternionic M\"obius group

An element of a group is called \emph{reversible} if it is conjugate to its inverse. While reversibility in the quaternionic M\"{o}bius group $\mathrm{PSL}(2,\mathbb{H})$ has traditionally been studied using geometric and dynamical methods, we develop a purely algebraic approach. We obtain an explicit, computable criterion for the reversibility of a quaternionic M\"{o}bius transformation, expressed solely in terms of the entries of a matrix representative. More precisely, we prove that \[ [A]\in \mathrm{PSL}(2,\mathbb{H}) \text{ is reversible} \quad \Longleftrightarrow \quad \beta_A^{2}=\delta_A^{2}, \] where $\beta_A$ and $\delta_A$ are real conjugacy invariants associated with a lift $A\in \mathrm{SL}(2,\mathbb{H})$. Furthermore, we give a complete characterization of reversing symmetries of reversible elements in $\mathrm{SL}(2,\mathbb{H})$ and $\mathrm{PSL}(2,\mathbb{H})$.

math.GT↗

arXiv2023

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

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 $\mathrm{SL}(n,\mathbb{C})$.

math.GR↗

arXiv2023

Reversibility and Real Adjoint Orbits of Linear Maps

We extend classical results on the classification of reversible elements of the group $\mathrm{GL}(n, \mathbb{C})$ (and $\mathrm{GL}(n, \mathbb{R})$) to $\mathrm{GL}(n, \mathbb{H})$ using an infinitesimal version of the classical reversibility, namely adjoint reality in the Lie algebra set-up. We also provide a new proof of such a classification for the general linear groups over $\mathbb{R}$ and $\mathbb{C}$. Further, we classify the real adjoint orbits in the Lie algebra $\mathfrak{gl}(n, \mathbb{D})$ for $ \mathbb{D}=\mathbb{R}, \mathbb{C}$ or $\mathbb{H} $.

math.GR↗

arXiv2022

Classification and Decomposition of Quaternionic Projective Transformations

We consider the projective linear group $\mathrm{PSL}(3,\mathbb{H})$. We have investigated the reversibility problem in this group and use the reversibility to offer an algebraic characterization of the dynamical types of $\mathrm{PSL}(3,\mathbb{H})$. We further decompose elements of $\mathrm{SL}(3,\mathbb{H})$ as products of simple elements, where an element $g$ in $\mathrm{SL}(3,\mathbb{H})$ is called $\textit{simple}$ if it is conjugate to an element of $\mathrm{SL}(3,\mathbb{R})$. We have also revisited real projective transformations and following Goldman's ideas, have offered a complete classification for elements of $\mathrm{SL }(3,\mathbb{R})$.

math.GR↗

arXiv2022

Reversibility of Affine Transformations

An element $g$ in a group $G$ is called reversible if $g$ is conjugate to $g^{-1}$ in $ G $. An element $g$ in $G$ is strongly reversible if $ g $ is conjugate to $g^{-1}$ by an involution in $G$. The group of affine transformations of $\mathbb{D}^n$ may be identified with the semi-direct product $\mathrm{GL}(n, \mathbb{D}) \ltimes \mathbb{D}^n $, where $\mathbb{D}:=\mathbb{R}, \mathbb{C}$ or $ \mathbb{H} $. This paper classifies reversible and strongly reversible elements in the affine group $\mathrm{GL}(n, \mathbb{D}) \ltimes \mathbb{D}^n $.

math.GR↗

arXiv2022

Limit sets of cyclic quaternionic Kleinian groups

In this paper, we consider the natural action of $\mathrm{SL}(3, \mathbb{H})$ on the quaternionic projective space $ \mathbb{P}_{\mathbb{H}}^2$. Under this action, we investigate limit sets for cyclic subgroups of $\mathrm{SL}(3, \mathbb{H})$. We compute two types of limit sets, which were introduced by Kulkarni and Conze-Guivarc'h, respectively.

math.GR↗

arXiv2022

Real adjoint orbits of special linear groups

Let $ G $ be a Lie group with Lie algebra $ \mathfrak{g} $. An element $ X \in \mathfrak{g} $ is called $\mathrm{Ad}_G$-real if $ -X=gXg^{-1} $ for some $ g \in G $. Moreover, if $ -X=gXg^{-1} $ holds for some involution $ g\in G $, then $ X $ is called strongly $\mathrm{Ad}_G$-real. We have classified the $\mathrm{Ad}_G$-real and the strongly $\mathrm{Ad}_G$-real orbits in the special linear Lie algebra $\mathfrak{sl}(n,\mathbb{F}) $ for $ \mathbb{F}=\mathbb{C}$ or $\mathbb{H} $.

math.GR↗

arXiv2022

Reversibility of Hermitian Isometries

An element $g$ in a group $G$ is called reversible (or real) if it is conjugate to $g^{-1}$ in $G$, i.e., there exists $h$ in $G$ such that $g^{-1}=hgh^{-1}$. The element $g$ is called strongly reversible if the conjugating element $h$ is an involution (i.e., element of order at most two) in $G$. In this paper, we classify reversible and strongly reversible elements in the isometry groups of $\mathbb{F}$-Hermitian spaces, where $\mathbb{F}=\mathbb{C}$ or $\mathbb{H}$. More precisely, we classify reversible and strongly reversible elements in the groups $ \mathrm{Sp}(n) \ltimes \mathbb{H}^n$, $\mathrm{U}(n) \ltimes \mathbb{C}^n$ and $\mathrm{SU}(n) \ltimes \mathbb{C}^n$. We also give a new proof of the classification of strongly reversible elements in $\mathrm{Sp}(n)$.

math.GR↗