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Ram Karan Choudhary

Publications and source records attributed to Ram Karan Choudhary.

10 recordsLinked to original sources
arXiv2026

On rational representations and rational group algebra of $\operatorname{GL}_2(q)$

In this article, we study rational representations of $G=\operatorname{GL}_2(q)$, where $q$ is a prime power. Let $ρ$ be an irreducible representation of $G$ over $\mathbb{Q}$. Then $ρ$ affords the character \[ Ω(χ)=m_{\mathbb{Q}}(χ)\sum_{σ\in\operatorname{Gal}(\mathbb{Q}(χ)/\mathbb{Q})}χ^σ, \] for some irreducible complex character $χ$ of $G$, where $m_{\mathbb{Q}}(χ)$ denotes the Schur index of $χ$ over $\mathbb{Q}$, and conversely, every character of this form is afforded by an irreducible representation of $G$ over $\mathbb{Q}$. We obtain a combinatorial description for the counting of inequivalent irreducible $\mathbb{Q}$-representations of $G$ of each distinct degree. Furthermore, we briefly determine the rational character table of $G$ and present a method for constructing an irreducible rational matrix representation $ρ$ of $G$ affording the character $Ω(χ)$, where $χ$ is an irreducible complex character of $G$ arising from parabolic induction. Finally, using the results on the rational representations of $G$, we derive an explicit combinatorial formula, depending only on $q$, for the Wedderburn decomposition of $\mathbb{Q}G$.

math.RT↗
arXiv2026

Waring Problem for matrices over finite local rings

This paper addresses the matrix Waring problem for matrices over finite principal local rings. Let $\mathcal{O}_{\ell}$ be a finite principal local ring of length $\ell$ with the maximal ideal $\mathfrak{m}$ and the residue field $\mathbb{F}_q = \mathcal{O}_\ell/\mathfrak{m}$. When $-1$ is a $k$-th power in $\mathbb{F}_q$ and the characteristic of $\mathbb{F}_q$ does not divide $k$, we show that for sufficiently large $q$, any matrix in $M_n(\mathcal{O}_\ell)$ can be expressed as a sum of two $k$-th powers. Furthermore, we establish that these two conditions are strictly necessary for the result to hold in general.

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arXiv2026

Wedderburn decomposition of the rational group algebras of $\operatorname{SL}_2(q)$ and $\operatorname{PSL}_2(q)$

In this article, we derive explicit combinatorial formulas, depending only on $q$, for the Wedderburn decomposition of the rational group algebras of the finite linear groups $\operatorname{SL}_2(q)$ and $\operatorname{PSL}_2(q)$. Furthermore, we also determine the number of pairwise non-isomorphic simple $\mathbb Q G$-modules of each possible dimension for $G$ being either $\operatorname{SL}_2(q)$ or $\operatorname{PSL}_2(q)$.

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arXiv2026

Classification of GVZ and Nested GVZ $p$-groups up to Order $p^6$

Let $G$ be a finite group and let $\Irr(G)$ denote the set of irreducible complex characters of $G$. For a normal subgroup $N \trianglelefteq G$ and $χ\in \Irr(G)$, we say that $χ$ is \emph{fully ramified} over $N$ if $χ(g)=0$ for all $g \in G \setminus N$. A group $G$ is said to be of \emph{central type} if there exists $χ\in \Irr(G)$ that is fully ramified over $Z(G)$. Motivated by this notion, an irreducible character $χ\in \Irr(G)$ is called of \emph{central type} if $χ$ vanishes on $G \setminus Z(χ)$, where \[ Z(χ)=\{\, g \in G : |χ(g)|=χ(1) \,\} \] is the center of $χ$. Groups in which every irreducible character is of central type are called \emph{GVZ-groups}. Furthermore, a group $G$ is said to be \emph{nested} if for all $χ,ψ\in \Irr(G)$, either $Z(χ)\subseteq Z(ψ)$ or $Z(ψ)\subseteq Z(χ)$. It is known that a GVZ-group is nilpotent. In this article, we classify all GVZ and nested GVZ $p$-groups of order at most $p^6$, where $p$ is an odd prime.

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arXiv2025

A Combinatorial Technique for the Wedderburn Decomposition of Rational Group Algebras of Nested GVZ $p$-groups

In this article, we present a combinatorial formula for the Wedderburn decomposition of rational group algebras of nested GVZ $p$-groups, where $p$ is an odd prime. Using this formula, we derive an explicit combinatorial expression for the Wedderburn decomposition of rational group algebras of all two-generator $p$-groups of class $2$. Additionally, we provide explicit combinatorial formulas for the Wedderburn decomposition of rational group algebras of certain families of nested GVZ $p$-groups with arbitrarily large nilpotency class. We also classify all nested GVZ $p$-groups of order at most $p^5$ and compute the Wedderburn decomposition of their rational group algebras. Finally, we determine a complete set of primitive central idempotents for the rational group algebras of nested GVZ $p$-groups.

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arXiv2025

On Matrix Representations of Groups of Order $p^5$ over $\mathbb{Q}$

In this article, we determine all inequivalent irreducible rational matrix representations of groups of order $p^5$, where $p$ is an odd prime. We also derive combinatorial formulations for the Wedderburn decomposition of rational group algebras of these $p$-groups, using results from their rational representations.

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arXiv2025

Rational Group Algebras of Camina $p$-groups

In this article, we present a combinatorial formula for the Wedderburn decomposition of rational group algebras of Camina $p$-groups, where $p$ is a prime. We also provide a complete set of primitive central idempotents of rational group algebras of these groups.

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arXiv2024

A Combinatorial Formula for the Wedderburn Decomposition of Rational Group Algebras and the Rational Representations of Ordinary Metacyclic $p$-groups

In this article, we present a combinatorial formula for computing the Wedderburn decomposition of the rational group algebra associated with an ordinary metacyclic $p$-group $G$, where $p$ is any prime. We also provide a formula for counting irreducible rational representations of $G$ with distinct degrees and derive a method to explicitly obtain all inequivalent irreducible rational matrix representations of $G$.

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arXiv2023

Rational Representations and Rational Group Algebra of VZ p-groups

In this article, we study rational matrix representations of VZ $p$-groups ($p$ is any prime). Utilizing our findings on VZ $p$-groups, we explicitly obtain all inequivalent irreducible rational matrix representations of all $p$-groups of order $\leq p^4$. Furthermore, we establish combinatorial formulas to determine the Wedderburn decompositions of rational group algebras for VZ $p$-groups and all $p$-groups of order $\leq p^4$, ensuring simplicity in the process.

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