arXiv subjects

Rijubrata Kundu

Publications and source records attributed to Rijubrata Kundu.

12 recordsLinked to original sources

arXiv2026

On Conjugacy Classes of Derangements in Symmetric and Alternating Groups

In this article, we prove two conjectures of Burness and Fusari [Timothy Burness and Marco Fusari, On derangements in simple permutation groups, Forum Math. Sigma 13 (2025)] concerning the powers and products of conjugacy classes of derangements in the symmetric and alternating groups: (1) We show that there exist two conjugacy classes $C$ and $D$ of derangements in $S_n$ such that $S_n=C^2\cup CD$, and (2) We show that there exists a conjugacy class $C$ of derangements in $A_n$ such that $C^2=A_n$, whenever $n\equiv 3\;(\text{mod}\;4)$. In fact, our result concerning the second conjecture holds in a considerably more general setting, which also answers affirmatively a question posed by Bertram [Edward Bertram, Even permutations as a product of two conjugate cycles, J. Comb. Theory, Ser. A 12 (1972), 368-380] in a particular case. Moreover, we show that any conjugacy class $C$ of derangements in $S_n$ (resp. $A_n$) contains a pair of elements that generate $S_n$ or $A_n$ (resp. $A_n$), unless $C$ is the conjugacy class of fixed-point-free involutions.

math.GR↗

arXiv2025

A Relationship Between Character Values Of Wreath Products And The Symmetric Group

A relation between certain irreducible character values of the hyperoctahedral group $B_n$ ($\mathbb{Z}/2\mathbb{Z} \wr S_n$) and the symmetric group $S_{2n}$ was proved by F. Lübeck and D. Prasad in 2021. Their proof is algebraic in nature and uses Lie theory. Using combinatorial methods, R. Adin and Y. Roichman proved a similar relation between certain character values of $G\wr S_n$ and $S_{rn}$, where $G$ is an abelian group of order $r$ (generalizing the result of Lübeck-Prasad). Using their result, we prove yet another relation between certain irreducible character values of $G\wr S_n$ and $S_{rn}$, where $G$ is an abelian group of order $r$.

math.CO↗

arXiv2024

Covering Numbers of Some Irreducible Characters of the Symmetric Group

The covering number of a non-linear character $χ$ of a finite group $G$ is the least positive integer $k$ such that every irreducible character of $G$ occurs in $χ^k$. We determine the covering numbers of irreducible characters of the symmetric group $S_n$ indexed by certain two-row partitions (and their conjugates), namely $(n-2,2)$ and $((n+1)/2, (n-1)/2)$ when $n$ is odd. We also determine the covering numbers of irreducible characters indexed by certain hook-partitions (and their conjugates), namely $(n-2,1^2)$, the almost self-conjugate hooks $(n/2+1, 1^{n/2-1})$ when $n$ is even, and the self-conjugate hooks $((n+1)/2, 1^{(n-1)/2})$ when $n$ is odd.

math.RT↗

arXiv2023

Products of conjugacy classes in $\text{SL}_2(k)$ and $\text{PSL}_2(k)$

Let $k$ be a field with $u$-invariant $\leq2$. Assume further that $k$ is not quadratically closed, $\mathsf{char}(k)\neq 2$ and $|k|\geq 5$. It is known that the covering number of both $\text{SL}_2(k)$ and $\text{PSL}_2(k)$ is three, while their extended covering number is four. In this article, we completely describe the product of two conjugacy classes in $\text{SL}_2(k)$ and $\text{PSL}_2(k)$. Further, we also describe the product of three conjugacy classes (at least two of which are distinct) in $\text{SL}_2(k)$ and $\text{PSL}_2(k)$.

math.GR↗

arXiv2022

Alternating groups as products of cycle classes - II

Given integers $k,l\geq 2$, where either $l$ is odd or $k$ is even, let $n(k,l)$ denote the largest integer $n$ such that each element of $A_n$ is a product of $k$ many $l$-cycles. In 2008, M. Herzog, G. Kaplan and A. Lev conjectured that $\lfloor \frac{2kl}{3} \rfloor \leq n(k,l)\leq \lfloor \frac{2kl}{3}\rfloor+1$. It is known that the conjecture holds when $k=2,3,4$. Moreover, it is also true when $3\mid l$. In this article, we determine the exact value of $n(k,l)$ when $3\nmid l$ and $k\geq 5$. As an immediate consequence, we get that $n(k,l)<\lfloor \frac{2kl}{3}\rfloor$ when $k\geq 5$, which shows that the above conjecture is not true in general. In fact, the difference between the exact value of $n(k,l)$ and the conjectured value grows linearly in terms of $k$. Our results also generalize the case of $k=2,3,4$.

math.CO↗

arXiv2022

Nilpotent Lie algebras with two centralizer dimensions over a finite field

A result of Barnea and Isaacs states that if $L$ is a finite dimensional nilpotent Lie algebra with exactly two distinct centralizer dimensions, then nilpotency class of $L$ is either $2$ or $3$. In this article, we classify all such finite dimensional $3$-step nilpotent Lie algebras over a finite field.

math.RA↗

arXiv2022

Alternating groups as products of cycle classes

Given integers $k,l\geq 2$, where either $l$ is odd or $k$ is even, let $n(k,l)$ denote the largest integer $n$ such that each element of $A_n$ is a product of $k$ many $l$-cycles. In 2008, M. Herzog, G. Kaplan and A. Lev proved that if $k,l$ both are odd, $3\mid l$ and $l>3$, then $n(k,l)=\frac{2}{3}kl$. They further conjectured that if $k$ is even and $3\mid l$, then $n(k,l)=\frac{2}{3}kl+1$. In this article, we prove this conjecture. We also prove that $n(k,3)=2k+1$ if $k$ is odd.

math.CO↗

arXiv2022

Counterexamples to a conjecture of M. Pellegrini and P. Shumyatsky

In this article, we provide counterexamples to a conjecture of M. Pellegrini and P. Shumyatsky which states that each coset of the centralizer of an involution in a finite non-abelian simple group $G$ contains an odd order element, unless $G=\text{PSL}(n,2)$ for $n\geq 4$. More precisely, we show that the conjecture does not hold for the alternating group $A_{8n}$ for all $n\geq 2$.

math.GR↗

arXiv2022

Generating functions for the powers in $\text{GL}(n,q)$

Consider the set of all powers $\text{GL}(n ,q)^M = \{x^M \mid x\in \text{GL}(n, q)\}$ for an integer $M\geq 2$. In this article, we aim to enumerate the regular, regular semisimple and semisimple elements as well as conjugacy classes in the set $\text{GL}(n, q)^M$, i.e., the elements or classes of these kinds which are $M^{th}$ powers. We get the generating functions for (i) regular and regular semisimple elements (and classes) when $(q,M)=1$, (ii) for semisimple elements and all elements (and classes) when $M$ is a prime power and $(q,M)=1$, and (iii) for all kinds when $M$ is a prime and $q$ is a power of $M$.

math.GR↗

arXiv2021

Nilpotent Lie Algebras of breadth type $(0,3)$

For a natural number $m$, a Lie algebra $L$ over a field $k$ is said to be of breadth type $(0, m)$ if the co-dimension of the centralizer of every non-central element is of dimension $m$. In this article, we classify finite dimensional nilpotent Lie algebras of breadth type $(0, 3)$ over $\mathbb F_q$ of odd characteristics up to isomorphism. We also give a partial classification of the same over finite fields of even characteristic, $\mathbb C$ and $\mathbb R$. We also discuss $2$-step nilpotent Camina Lie algebras.

math.RA↗

arXiv2020

Powers in the wreath product of $G$ with $S_n$

In this paper we compute powers in the wreath product $G\wr S_n$, for any finite group $G$. For $r\geq 2$, a prime, consider $ω_r: G\wr S_n\to G\wr S_n$ defined by $g \mapsto g^r$. Let $P_{r}(G\wr S_n)=\frac{|ω_r(G\wr S_n)|}{|G|^n n!}$, be the probability that a randomly chosen element in $G\wr S_n$ is a $r^{th}$ power. We prove, $P_r(G\wr S_{n+1})=P_r(G\wr S_n)$ for all $n\not \equiv -1(\text{mod } r)$ if, order of $G$ is coprime to $r$. We also give a formula for the number of conjugacy classes that are $r^{th}$ powers in $G\wr S_n$.

math.GR↗

arXiv2020

Asymptotics of the powers in finite reductive groups

Let $G$ be a connected reductive group defined over $\mathbb F_q$. Fix an integer $M\geq 2$, and consider the power map $x\mapsto x^M$ on $G$. We denote the image of $G(\mathbb F_q)$ under this map by $G(\mathbb F_q)^M$ and estimate what proportion of regular semisimple, semisimple and regular elements of $G(\mathbb F_q)$ it contains. We prove that as $q\to\infty$, all of these proportions are equal and provide a formula for the same. We also calculate this more explicitly for the groups $\text{GL}(n,q)$ and $\text{U}(n,q)$.

math.GR↗