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
- Lie Groups & Algebras
- Algebraic Groups
- Chevalley Groups
- Representation Theory
List of Publications
Preprints
-
6.
Automorphisms of Twisted Chevalley Groups over Commutative Rings arXiv
We classify all abstract automorphisms of the elementary and full twisted Chevalley groups of types ${}^2A_\ell$ ($\ell\geq3$), ${}^2D_\ell$ ($\ell\geq4$), ${}^2E_6$, and ${}^3D_4$ over arbitrary commutative rings, for every invariant isogeny type. The coefficient automorphism $\theta$ has order dividing that of the diagram automorphism. We assume that $2$ is invertible and, for triality, that $3$ is invertible as well.
Three features distinguish the answer from the usual inner--diagonal and coefficient decomposition. The elementary groups of types ${}^2A_3$ and ${}^2A_4$ admit an additive family of exceptional automorphisms supported at $\theta$-fixed maximal ideals with residue field $\mathbb F_3$. In type ${}^2D_4$, diagram automorphisms may vary on idempotent components, subject to compatibility with the involution and the isogeny lattice. In triality, the coefficient automorphisms preserve the trace operator $r\mapsto r+\theta(r)+\theta^2(r)$ but need not commute with $\theta$. We determine which elementary automorphisms extend to the full groups and describe the remaining central factors. A nonzero exceptional unitary factor extends precisely in the simply connected type ${}^2A_3$, giving a classification for the hyperbolic group $\operatorname{SU}_4(R,\theta;Q_4)$. The proof glues uniquely determined normalization data for a finite root frame and then recovers the coefficient action from root-subgroup relations.
-
5.
Automorphisms of triality groups over local rings arXiv
Let \(R\) be a nonzero commutative local ring with \(2,3\in R^\times\), and let \(\rho\in\operatorname{Aut}(R)\) satisfy \(\rho^3=\mathrm{id}\). We classify abstract automorphisms of the elementary and full triality groups in the adjoint and simply connected forms of type \(D_4\). Every automorphism is uniquely determined by an adjoint group element and a ring automorphism preserving the trace \(T=\mathrm{id}+\rho+\rho^2\); conversely, every such pair defines an automorphism. A square-zero example shows that trace-preserving coefficient automorphisms need not normalize \(\langle\rho\rangle\) when \(\rho\) is nontrivial on \(R\) but trivial on its residue field. We also prove a relative rigidity theorem for homomorphisms from elementary triality groups over arbitrary coefficient rings to local targets with \(2\) and \(3\) invertible, assuming that an adjoint conjugation makes the parameter-one root images standard modulo the maximal ideal. Conversely, every unital trace-preserving ring homomorphism into such a local coefficient ring induces the corresponding homomorphism of elementary groups.
-
4.
Automorphisms of twisted Chevalley groups of type \(^2E_6\) over local rings arXiv
We classify abstract automorphisms of the elementary and full twisted Chevalley groups of type \(^2E_6\) over commutative local rings with involution and \(2\) invertible, for both isogeny types. Every automorphism is standard, with no invertibility assumption on \(3\). In the adjoint case, it has a unique expression as a composition of conjugation by an element of the full twisted group and a ring automorphism commuting with the involution. For the simply connected form, the full group equals its elementary subgroup, and the factors are strictly inner, diagonal, and ring automorphisms. The main step replaces a linear change of basis by a group element. Invariant alternating tensors determine the transported Chevalley bracket. We correct this bracket over a faithfully flat local extension and descend the resulting conjugating element by uniqueness. Root-subgroup relations then recover the coefficient automorphism.
-
3.
Automorphisms of twisted Chevalley groups of type \(^2D_\ell\ (\ell \ge 4)\) over local rings arXiv
We prove that all automorphisms of twisted Chevalley groups of type \(^2D_\ell\), \(\ell\geq 4\), and their elementary subgroups over commutative local rings with \(1/2\) are standard. More precisely, for elementary groups, every automorphism is a composition of strictly inner, diagonal, and ring automorphisms; for full groups, a central factor may also occur. In the adjoint case, diagonal automorphisms are strictly inner and the centre is trivial, so only strictly inner and ring factors are needed for both the elementary and full groups.
-
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.
-
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
-
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.
-
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 \(G_{\pi,\sigma}(\Phi, R)\).
-
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
Here I provide an overview of my research, organized project by project.
-
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.
The classification over arbitrary commutative rings is now available in our preprint Automorphisms of Twisted Chevalley Groups over Commutative Rings, for every invariant isogeny type, assuming that \(2\) is invertible and, in triality, \(3\) is invertible as well.
-
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.
-
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 \(E'_\sigma(R)\).