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Renu Joshi

Publications and source records attributed to Renu Joshi.

6 recordsLinked to original sources
arXiv2024

Dualizing involutions on the $n$-fold metaplectic cover of $\GL(2)$

Let $F$ be a non-Archimedean local field of characteristic zero and $G=\GL(2,F)$. Let $n\geq 2$ be a positive integer and $\widetilde{G}=\widetilde{\GL}(2,F)$ be the $n$-fold metaplectic cover of $G$. Let $\pi$ be an irreducible smooth representation of $G$ and $\pi^{\vee}$ be the contragredient of $\pi$. Let $\tau$ be an involutive anti-automorphism of $G$ satisfying $\pi^{\tau}\simeq \pi^{\vee}$. In this case, we say that $\tau$ is a dualizing involution. A well known theorem of Gelfand and Kazhdan says that the standard involution $\tau$ on $G$ is a dualizing involution. In this paper, we show that any lift of the standard involution to $\widetilde{G}$ is a dualizing involution if and only if $n=2$.

math.RT↗
arXiv2024

Combinatorial invariants for certain classes of non-abelian groups

This article focuses on the study of zero-sum invariants of finite non-abelian groups. We address two main problems: the first centers on the ordered Davenport constant and the second on Gao's constant. We establish a connection between the ordered Davenport constant and the small Davenport constant for a finite non-abelian group of even order, which in turn gives a relation with the Noether number. Additionally, we confirm a conjecture of Gao and Li for a non-abelian group of order $2p^{\alpha}$, where $p$ is a prime. Furthermore, we prove a conjecture that connects the ordered Davenport constant to the Loewy length for certain classes of finite $2$-groups.

math.CO↗
arXiv2024

An algebraic approach towards a conjecture on the Davenport constant

For a finite group $G,$ $\mathsf{D}(G)$ is defined as the least positive integer $k$ such that for every sequence $S=g_1\bdot g_2\bdot \dotsc \bdot g_k$ of length $k$ over $G$, there exist $1 \le i_1 < i_2 <\cdots < i_m \le k $ such that $g_{i_1}g_{i_2}\cdots g_{i_m}=1,$ where $1$ is the identity element of $G.$ The small Davenport constant $\mathsf{d}(G)$ is the maximal positive integer $k$ such that there is a sequence of length $k$ over $G$ which has no non-trivial product-one subsequence. In 2004, Dimitrov proved that $\mathsf{D}(G)\leq \mathsf{L}(G)$ for a finite $p$-group $G$, where $p$ is a prime and $\mathsf{L}(G)$ is the Loewy length of $\mathbb{F}_p[G].$ He conjectured that the equality holds for all finite $p$-groups. In this article, we compute $\mathsf{D}(G)$ for certain classes of finite non-abelian $p$-groups, including metacyclic groups, and show that the conjecture is true by determining the precise value of $\mathsf{L}(G)$. As a consequence, we refine an upper bound on $\mathsf{d}(G)$ recently given by Qu, Li and Teeuwsen, and prove that for specific classes of groups $\mathsf{D}(G)=\mathsf{d}(G)+1$. We also evaluate $\mathsf{D}(G)$ for finite dicyclic, semi-dihedral and other groups.

math.CO↗
arXiv2024

The Bogomolov multiplier of a multiplicative Lie algebra

In this paper, we develop the concept of the Bogomolov multiplier for a multiplicative Lie algebra and establish a Hopf-type formula. Consequently, we see that the Bogomolov multipliers of two isoclinic multiplicative Lie algebras are isomorphic.

math.GR↗
arXiv2020

A survey on the Non-inner Automorphism Conjecture

In this survey article, we try to summarize the known results towards the long-standing non-inner automorphism conjecture, which states that every finite non-abelian $p$-group has a non-inner automorphism of order $p$.

math.GR↗