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Deep H. Makadiya

Publications and source records attributed to Deep H. Makadiya.

7 recordsLinked to original sources
arXiv2026

Automorphisms of Twisted Chevalley Groups of type ${}^2 A_3$ and ${}^2 A_4$ over Local Rings

This paper is part of an ongoing series devoted to the classification of automorphisms and isomorphisms of twisted Chevalley groups over commutative rings. In our previous work~\cite{EB&DM1}, we established that over local rings containing $1/2$, all automorphisms of twisted Chevalley groups and their elementary subgroups of type ${}^2A_{\ell}$ ($\ell \geq 5$) are standard. In the present paper, we address the remaining low-rank cases and provide a complete classification of the automorphisms of groups of types ${}^2A_3$ and ${}^2A_4$ over local rings $R$ containing $1/2$, across all isogeny types.

math.GR↗
arXiv2025

On Normal Subgroups of Twisted Chevalley Groups over Commutative Rings

In this paper, we prove two structure theorems for twisted Chevalley groups $G_σ(R)$ over a commutative ring $R$ with unity. The first theorem concerns the normality of $E'_σ(R,J)$, the elementary congruence subgroups at level $J$, in the group $G_σ(R)$. The second theorem classifies all subgroups of $G_σ(R)$ normalized by its elementary subgroup $E'_σ(R)$. Along the way, we obtain several interesting results. For instance, when $R$ is a semilocal ring, we show that $G_σ(R)$ can be expressed as the (internal) product of $E'_σ(R)$ and the maximal torus $T_σ(R)$ of $G_σ(R)$.

math.GR↗
arXiv2025

Some Properties of Twisted Chevalley Groups

This thesis investigates certain structural properties of twisted Chevalley groups over commutative rings, focusing on three key problems. Let $R$ be a commutative ring satisfying mild conditions. Let $G_{π,σ} (Φ, R)$ denote a twisted Chevalley group over $R$, and let $E'_{π, σ} (Φ, R)$ denote its elementary subgroup. The first problem concerns the normality of $E'_{π, σ} (Φ, R, J)$, the relative elementary subgroups at level $J$, in the group $G_{π, σ} (Φ, R)$. The second problem addresses the classification of the subgroups of $G_{π, σ}(Φ, R)$ that are normalized by $E'_{π, σ}(Φ, R)$. This classification provides a comprehensive characterization of the normal subgroups of $E'_{π, σ}(Φ, R)$. Lastly, the third problem investigates the normalizers of $E'_{π, σ}(Φ, R)$ and $G_{π, σ}(Φ, R)$ in the bigger group $G_{π, σ}(Φ, S)$, where $S$ is a ring extension of $R$. We prove that these normalizers coincide. Moreover, for groups of adjoint type, we show that they are precisely equal to $G_{π, σ}(Φ, R)$.

math.GR↗
arXiv2025

Triangular and Unitriangular Factorization of Twisted Chevalley Groups

The existence of triangular and unitriangular factorizations has been extensively studied for untwisted Chevalley groups, as well as for twisted Chevalley groups of types other than ${}^2A_{2n} \ (n \geq 1)$. However, the case of twisted Chevalley groups of type ${}^2A_{2n} \ (n \geq 1)$, has remained unresolved in the general setting of commutative rings. Prior work by A. Smolensky addressed this case only over certain fields, including finite fields and the field of complex numbers. These results indicate that, even over fields, the ${}^2A_{2n}$ case demands more refined techniques, reflecting the difficulty of extending such factorizations to the broader class of commutative rings. In this paper, we introduce two new classes of commutative rings: those satisfying the \emph{special stable range one condition} and those that are \emph{$θ$-complete}. We discuss their basic properties and provide illustrative examples. Our main result establishes the existence of triangular and unitriangular factorizations for twisted Chevalley groups of type ${}^2A_{2n}$ over a certain class of commutative rings, which includes all fields, all local rings (with mild restrictions), and several other important classes of rings.

math.GR↗
arXiv2025

Normalizer of Twisted Chevalley Groups over Commutative Rings

Let $R$ be a commutative ring with unity. Consider the twisted Chevalley group $G_{π, σ} (Φ, R)$ of type $ϕ$ over $R$ and its elementary subgroup $E'_{π, σ} (Φ, R)$. This paper investigates the normalizers of $E'_{π, σ}(Φ, R)$ and $G_{π, σ}(Φ, R)$ in the larger group $G_{π, σ}(Φ, S)$, where $S$ is an extension ring of $R$. We establish that under certain conditions on $R$ these normalizers coincide. Moreover, in the case of adjoint type groups, we show that they are precisely equal to $G_{π, σ}(Φ, R)$.

math.GR↗