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Shripad M. Garge

Publications and source records attributed to Shripad M. Garge.

14 recordsLinked to original sources
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

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↗
arXiv2024

Effective cone of a Grassmann bundle over a curve defined over $\overline{\mathbb F}_p$

Let $X$ be an irreducible smooth projective curve defined over $\overline{\mathbb F}_p$ and $E$ a vector bundle on $X$ of rank at least two. For any $1\, \leq\, r\, <\, {\rm rank}(E)$, let ${\rm Gr}_r(E)$ be the Grassmann bundle over $X$ parametrizing all the $r$ dimensional quotients of the fibers of $E$. We prove that the effective cone in ${\rm NS}({\rm Gr}_r(E))\otimes_{\mathbb Z} {\mathbb R}$ coincides with the pseudo-effective cone in ${\rm NS}({\rm Gr}_r(E))\otimes_{\mathbb Z} {\mathbb R}$. When $r\,=\,1$ or ${\rm rank}(E)-1$, this was proved by A. Moriwaki.

math.AG↗
arXiv2023

Torus Quotients of Richardson Varieties

For $1\le r\le n-1,$ let $G_{r,n}$ denote the Grassmannian parametrizing $r$-dimensional subspaces of $\mathbb{C}^{n}.$ Let $(r,n)=1.$ In this article we show that the GIT quotients of certain Richardson varieties in $G_{r,n}$ for the action of a maximal torus in $SL(n,\mathbb{C})$ are the product of projective spaces with respect to the descent of a suitable line bundle.

math.AG↗
arXiv2022

Seshadri Constants Over Fields Of Characteristic Zero

Let $X$ be a smooth projective variety defined over a field $k$ of characteristic $0$ and let $\mathcal{L}$ be a nef line bundle defined over $k$. We prove that if $x\in X$ is a $k$-rational point then the Seshadri constant $ε(X, \mathcal{L}, x)$ over $\overline{k}$ is the same as that over $k$. We show, by constructing families of examples, that there are varieties whose global Seshadri constant $ε(X)$ is zero. We also prove a result on the existence of a Seshadri curve with a natural (and necessary) hypothesis.

math.AG↗
arXiv2021

Asymptotics of commuting probabilities in reductive algebraic groups

Let $G$ be an algebraic group. For $d\geq 1$, we define the commuting probabilities $cp_d(G) = \frac{dim(\mathfrak C_d(G))}{dim(G^d)}$, where $\mathfrak C_d(G)$ is the variety of commuting $d$-tuples in $G$. We prove that for a reductive group $G$ when $d$ is large, $cp_d(G)\sim \fracα{n}$ where $n=\dim(G)$, and $α$ is the maximal dimension of an Abelian subgroup of $G$. For a finite reductive group $G$ defined over the field $\mathbb F_q$, we show that $cp_{d+1}(G(\mathbb F_q))\sim q^{(α-n)d}$, and give several examples.

math.GR↗
arXiv2021

Commuting probability in algebraic groups

We introduce the notion of commuting probability, $p(G)$, for an algebraic group $G$. This notion is inspired by the corresponding notions in finite groups and compact groups. The computation of $p(G)$ for reductive groups is readily done using the notion of $z$-classes. We introduce two generalisations of this relation, $iz$-equivalence and $dz$-equivalence. These notions lead us naturally to the notion of a regular element in $G$. Finally, with the help of this notion of regular elements, we compute $p(G)$ for a connected, linear algebraic group $G$. We also compute the set of limit points of the numbers $p(G)$ as $G$ varies over the classes of reductive groups, solvable groups and nilpotent groups.

math.GR↗
arXiv2020

Finiteness of $z$-classes in reductive groups

Let $k$ be a perfect field such that for every $n$ there are only finitely many field extensions, up to isomorphism, of $k$ of degree $n$. If $G$ is a reductive algebraic group defined over $k$, whose characteristic is very good for $G$, then we prove that $G(k)$ has only finitely many $z$-classes. For each perfect field $k$ which does not have the above finiteness property we show that there exist groups $G$ over $k$ such that $G(k)$ has infinitely many $z$-classes.

math.GR↗
arXiv2009

On Gelfand models for finite Coxeter groups

A Gelfand model for a finite group $G$ is a complex linear representation of $G$ that contains each of its irreducible representations with multiplicity one. For a finite group $G$ with a faithful representation $V$, one constructs a representation which we call the polynomial model for $G$ associated to $V$. Araujo and others have proved that the polynomial models for certain irreducible Weyl groups associated to their canonical representations are Gelfand models. In this paper, we give an easier and uniform treatment for the study of the polynomial model for a general finite Coxeter group associated to its canonical representation. Our final result is that such a polynomial model for a finite Coxeter group $G$ is a Gelfand model if and only if $G$ has no direct factor of the type $W(D_{2n}), W(E_7)$ or $W(E_8)$.

math.GR↗
arXiv2006

On the order of finite semisimple groups

It is a theorem of Artin, Tits et al. that a finite simple group is determined by its order, with the exception of the groups (A_3(2), A_2(4)) and (B_n(q), C_n(q)) for n > 2, q odd. We investigate the situation for finite semisimple groups of Lie type. It turns out that the order of the finite group H(F_q) for a split semisimple algebraic group H defined over F_q, does not determine the group H upto isomorphism, but it determines the field F_q under some mild conditions. We then put a group structure on the pairs (H_1, H_2) of split semisimple groups defined over a fixed field F_q such that the orders of the finite groups H_1(F_q) and H_2(F_q) are the same and the groups H_i have no common simple direct factors. We obtain an explicit set of generators for this abelian, torsion-free group. We finally give a geometric reasoning for these order coincidences.

math.GR↗
arXiv2006

Maximal tori determining the algebraic group

Let k be a finite field, a global field or a local non-archimedean field. Let H_1 and H_2 be two split, connected, semisimple algebraic groups defined over k. We prove that if H_1 and H_2 share the same set of maximal k-tori up to k-isomorphism, then the Weyl groups W(H_1) and W(H_2) are isomorphic, and hence the algebraic groups modulo their centers are isomorphic except for a switch of a certain number of factors of type B_n and C_n. We remark that due to a recent result of Philippe Gille, above result holds for fields which admit arbitrary cyclic extensions.

math.GR↗
arXiv2004

Arithmetic of algebraic groups

This thesis studies arithmetic of linear algebraic groups. It involves studying the properties of linear algebraic groups defined over global fields, local fields and finite fields, or more generally the study of the linear algebraic groups defined over the fields which admit arbitrary cyclic extensions.

math.GR↗