Research
My areas of research interest, a list of publications, and an overview of the research projects I am currently working on or have worked on.
Areas of Interest
- Algebraic Groups
- Chevalley Groups & Algebras
- Lie Groups & Algebras
- Representation Theory
List of Publications
Preprints
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6.
Automorphisms of twisted Chevalley groups over commutative rings
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5.
Automorphisms of twisted Chevalley groups of type \(^3D_4\) over local rings
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4.
Automorphisms of twisted Chevalley groups of type \(^2E_6\) over local rings
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3.
Automorphisms of twisted Chevalley groups of type \(^2D_\ell\ (\ell \ge 4)\) over local rings arXiv
This paper is part of a series devoted to the classification of isomorphisms and automorphisms of twisted Chevalley groups over commutative rings. In this work, we prove that every automorphism of a twisted Chevalley group of type \(^2D_\ell\ (\ell \geqslant 4)\) over a local ring containing \(1/2\) is standard.
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2.
Automorphisms of twisted Chevalley groups of type \(^2A_3\) and \(^2A_4\) over local rings arXiv
We give a complete classification of the automorphisms of twisted Chevalley groups of types ${}^2A_3$ and ${}^2A_4$, and of their elementary subgroups, over commutative local rings $R$ containing $1/2$ and equipped with a non-trivial involution $\theta$. The classification covers all isogeny types and, together with our higher-rank result, completes the description for type ${}^2A_{\ell}$ with $\ell\geq3$.
Unlike the higher-rank groups, the elementary groups in these two types can have genuinely nonstandard automorphisms. Let $J$ be the Jacobson radical of $R$ and write $R^-_\theta=\{r\in R\mid\theta(r)=-r\}$. The exceptional family is parametrized by \[ \mathcal E(R,\theta)= \begin{cases} \operatorname{Ann}_R(J)\cap R^-_\theta, & \text{if }R/J\cong\mathbf F_3,\\[2mm] \{0\}, & \text{if }R/J\not\cong\mathbf F_3. \end{cases} \] For each $\delta\in\mathcal E(R,\theta)$ and $n\in\{3,4\}$, we explicitly construct an automorphism $\Theta_\delta^{(n)}$ of $E'_{\pi,\sigma}(A_n,R)$, which is nonstandard precisely when $\delta\neq0$. These automorphisms arise from explicit $1$-cocycles of the residual elementary group, lifted to a square-zero congruence layer over $R$. They account for all failures of standardness: every automorphism of the elementary group is a composition of a strictly inner automorphism, a diagonal automorphism, a ring automorphism commuting with $\theta$, and a single exceptional automorphism $\Theta_\delta^{(n)}$.
For the full groups, the answer depends on the isogeny type. A nontrivial exceptional automorphism extends to the full group precisely in the simply connected case of type ${}^2A_3$, where the full group is $\operatorname{SU}_4(R,Q_4)$ and coincides with its elementary subgroup. For every other isogeny type of ${}^2A_3$ and ${}^2A_4$, all automorphisms of the full group are standard: they are compositions of strictly inner, diagonal, ring, and central automorphisms.
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1.
Automorphisms of twisted Chevalley groups of type \(^2A_\ell\ (\ell \ge 5)\) over local rings arXiv
This paper is the first in a series devoted to the classification of automorphisms and isomorphisms of twisted Chevalley groups over commutative rings. In the present paper, we prove that every automorphism of a twisted Chevalley group of type \(^2A_\ell\ (\ell \ge 5)\) over a local ring in which \(2\) is invertible is standard.
Published & Accepted
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3.
Triangular and unitriangular factorization of twisted Chevalley groups arXiv
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 \(^2A_{2n}\ (n \ge 1)\). However, the case of twisted Chevalley groups of type \(^2A_{2n}\ (n \ge 1)\), 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 special stable range one condition and those that are \(\theta\)-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.
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2.
Normalizer of twisted Chevalley groups over commutative rings DOI arXiv
Let \(R\) be a commutative ring with unity. Consider the twisted Chevalley group \(G_{\pi,\sigma}(\Phi, R)\) of type \(\Phi\) over \(R\) and its elementary subgroup \(E'_{\pi,\sigma}(\Phi, R)\). This paper investigates the normalizers of \(E'_{\pi,\sigma}(\Phi, R)\) and \(G_{\pi,\sigma}(\Phi, R)\) in the larger group \(G_{\pi,\sigma}(\Phi, S)\), 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 \(G_{\pi,\sigma}(\Phi, R)\).
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1.
On normal subgroups of twisted Chevalley groups over commutative rings DOI arXiv
In this paper, we prove two structure theorems for twisted Chevalley groups \(G_\sigma(R)\) over a commutative ring \(R\) with unity. The first theorem concerns the normality of \(E'_\sigma(R, J)\), the elementary congruence subgroups at level \(J\), in the group \(G_\sigma(R)\). The second theorem classifies all subgroups of \(G_\sigma(R)\) normalized by its elementary subgroup \(E'_\sigma(R)\). Along the way, we obtain several interesting results. For instance, when \(R\) is a semilocal ring, we show that \(G_\sigma(R)\) can be expressed as the (internal) product of \(E'_\sigma(R)\) and the maximal torus \(T_\sigma(R)\) of \(G_\sigma(R)\).
Research Overview
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3.
Classification of the automorphisms of twisted Chevalley groups
The automorphisms of the classical groups and of the untwisted Chevalley groups over commutative rings are by now understood. For the twisted groups the description was known only over fields. With Prof. Elena I. Bunina I have begun a programme to classify all automorphisms of twisted Chevalley groups over commutative rings, for which my earlier work with Prof. Shripad M. Garge provides the groundwork.
We began with local rings. Across a series of papers we have shown that for types \(^2A_\ell\ (\ell \ge 5)\), \(^2D_\ell\ (\ell \ge 4)\), \(^2E_6\) and \(^3D_4\) over a local ring containing \(1/2\), every automorphism is standard. Types \(^2A_3\) and \(^2A_4\) are the exception: there an automorphism need not be standard. Precisely, a non-standard one exists when the residue field is \(\mathbf{F}_3\) and the skew-symmetric part of the annihilator of the Jacobson radical is non-zero. In every case the argument runs the same way — first for the elementary subgroup of adjoint type, then for the elementary subgroup attached to an arbitrary lattice, and finally for the full group.
The commutative ring case builds on the local one. Again we begin with the adjoint elementary group, where the picture matches what we found over local rings, and pass by the same route to the elementary subgroup for an arbitrary lattice. Getting to the full group requires knowing the normalizer, which is the subject of another of my projects with Prof. Garge: we settled it for rings containing \(1/2\) that also carry an invertible skew-symmetric element. However, we are now redoing the proof to drop the latter hypothesis.
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2.
Triangular and unitriangular factorization of twisted Chevalley groups
Whether a group is a bounded product of its upper and lower (uni)triangular subgroups is a basic structural question, settled long ago for the untwisted Chevalley groups and for every twisted type except \(^2A_{2n}\). That type is set aside so often for a reason that is structural rather than incidental: the twisting identifies two adjacent nodes of the \(A_{2n}\) diagram — an \(A_2\) subsystem — where in every other twisted type the identified nodes are orthogonal. The root subgroup this produces is not abelian, and the commutator calculus that carries the other types through has nothing to work with. A. Smolensky had reached the case over finite fields and over the complex numbers; over general commutative rings it stayed open.
Working with Prof. Shripad M. Garge, we introduced two classes of commutative rings — those satisfying a special stable range one condition, and those that are \(\theta\)-complete — and used them to establish triangular and unitriangular factorizations for type \(^2A_{2n}\) over a class of rings that takes in all fields, all local rings with mild restrictions, and several other familiar classes.
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1.
Sandwich classification of normal subgroups of twisted Chevalley groups
The sandwich classification of normal subgroups is a classical piece of the theory of classical and untwisted Chevalley groups over commutative rings. For the twisted groups the corresponding question was open in that generality.
With Prof. Shripad M. Garge I settled it in my doctoral work. In fact we prove a stronger statement, about subgroups normalized by the elementary subgroup rather than only the normal ones: every subgroup \(H\) of a twisted Chevalley group \(G_\sigma(R)\) normalized by the elementary subgroup \(E'_\sigma(R)\) determines a unique \(\theta\)-invariant ideal \(J\) of \(R\) with \(E'_\sigma(R,J) \subseteq H \subseteq G_\sigma(R,J)\), and conversely, every subgroup lying between those two is normalized by \(E'_\sigma(R)\).