arXiv subjects

Saikat Panja

Publications and source records attributed to Saikat Panja.

At least 19 recordsLinked to original sources

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 $\operatorname{PSL}_2(q)$.

math.RT↗

arXiv2026

Periodic Points of Power Maps in Finite Matrix Groups and Algebras

Consider the power map $x\mapsto x^L$ for a prime $L\neq 2$ such that $L|q-1$ where $q$ is a power of a prime. We determine the periodic points under this map for $\operatorname{M}_n(q)$, the algebra of $n\times n $ matrices over a finite field of order $q$, and also for the group $\operatorname{GL}_n(q)=\operatorname{M}_n(q)^\times$. We compute the limit $ \lim\limits_{\substack{q\longrightarrow \infty\\v_L(q-1)=c}}\dfrac{\left|\operatorname{Per}(x^L,\operatorname{M}_\ell(q))\right|}{|\operatorname{M}_\ell(q)|}$ and consequently $\lim\limits_{\substack{q\longrightarrow \infty v_L(q-1)=c}}\dfrac{\left|\operatorname{Per}(x^L,\operatorname{GL}_\ell(q))\right|}{|\operatorname{GL}_\ell(q)|}$, where $v_L$ denotes the $L$-adic valuation. We also compute the quantity $\lim\limits_{\substack{q\longrightarrow \infty v_L(q-1)=c}}\dfrac{\left|\operatorname{Per}(x^L,\operatorname{Sp}_{2\ell}(q))\right|}{|\operatorname{Sp}_{2\ell}(q)|}$ and $\lim\limits_{\substack{q\longrightarrow \infty v_L(q-1)=c}}\dfrac{\left|\operatorname{Per}(x^L,\operatorname{U}_\ell(q))\right|}{|\operatorname{U}_\ell(q)|}$; turns out these two limiting values are same. In all the cases, it turns out that the regular semisimple elements play the role in determining the limiting values.

math.NT↗

arXiv2026

Lifting solutions of polynomial equations on matrices over field to complete local principal ideal rings

Let $\widehat{\mathscr O}$ be a complete local principal ideal ring with residue field $k$ of characteristic not $2$ and $f\in \widehat{\mathscr O}[x_1,x_2,\dots,x_m]$. Take $A\in \mathrm M_n(\widehat{\mathscr O})$ with its reduction $\overline{A}\in \mathrm M_n(k)$. In this article, we study the following lifting problem. Suppose there exists a tuple $(\widetilde{B}_1, \widetilde{B}_2, \dots,\widetilde{B}_m)\in \mathrm M_n(k)^m$ of pairwise commuting matrices such that $f(\widetilde{B}_1, \widetilde{B}_2, \dots,\widetilde{B}_m) = \overline{A}$; under what conditions can this solution be lifted to a tuple $(B_1,B_2,\dots,B_m)\in \mathrm M_n(\widehat{\mathscr O})^m$ of pairwise commuting matrices satisfying $f(B_1,B_2,\dots,B_m)=A$? For $\overline{A}$ cyclic, we show that, under suitable hypotheses analogous to those appearing in Hensel lemma, such a lifting is always possible.

math.GR↗

arXiv2025

Dynamics of Word Maps on Groups and Polynomial Maps on Algebras

We introduce the notions of Fatou and Julia sets in the context of word maps on complex Lie groups and polynomial maps on finite-dimensional associative $\mathbb C$-algebras. For the group-theoretic question, we investigate the dynamics of the power map $x \mapsto x^{M}$ on the Lie group $\mathrm{GL}_n(\mathbb C)$, where $M \geq 2$ is an integer. For the algebra-related question, we study polynomial self-maps of $\mathrm{M}_n(\mathbb C)$ induced by monic polynomials in one variable. In both cases, we pin down the explicit description of the Fatou and Julia sets. We also show that there does not exist any wandering Fatou component of the pair $(p,\mathrm M_n(\mathbb C))$ where $p\in\mathbb C[z]$ is a monic polynomial of degree $\geq 2$.

math.DS↗

arXiv2025

Power maps on General Linear groups over finite principal ideal local rings of length two

Word maps have been studied for matrix groups over a field. We initiate the study of problems related to word maps in the context of the group $\mathrm{GL}_n(\mathscr O_2)$, where $\mathscr O_2$ is a finite local principal ideal ring of length two (e.g. $\mathbb{Z}/p^2\mathbb{Z}$ and $\mathbb F_q[t]/\langle t^2\rangle$). We study the power map $g\mapsto g^L$, where $L$ is a positive integer. We consider $L$ to be coprime to $p$ (an odd prime), the characteristic of the residue field $k$ of $\mathscr O_2$. We classify all the elements in the image, whose mod-$\mathfrak m$ reduction in $\mathrm{GL}_n(k)$ are either regular semisimple or cyclic, where $\mathfrak m$ is the unique maximal ideal of $\mathscr O_2$. Our main tool is a Hensel lifting for polynomial equations over $\mathrm{M}_n(\mathscr O_2)$, which we establish in this work. A central contribution of this work is the construction of canonical forms for certain natural classes of matrices over $\mathscr O_2$. As applications, we derive explicit generating functions for the probabilities that a random element of $\mathrm{GL}_n(\mathscr O_2)$ is regular semisimple, $L$-power regular semisimple, compatible cyclic, or $L$-power compatible cyclic.

math.GR↗

arXiv2025

Rational conjugacy classes and rational characters for some finite simple groups

If $G$ is a finite group, an irreducible complex-valued character $χ$ is called rational if $χ(g)$ is rational for all $g\in G$. Also, a conjugacy class $x^G$ is called rational, if for all irreducible complex-valued character $χ$, the value $χ(x^G)$ is rational. We prove that for $q$, a power of prime, the group $\mathrm{PSL}_2(q)$ has same number of rational characters and rational conjugacy classes. Furthermore, we verify that this equality holds for all finite simple groups whose character tables appear in the $\textit{ATLAS of Finite Groups}$, except for the Tits group.

math.GR↗

arXiv2025

Polynomial maps with constants on split octonion algebras

Let $\mathbf{O}(\mathbb{F})$ be the split octonion algebra over an algebraically closed field $\mathbb{F}$. For positive integers $k_1, k_2\geq 2$, we study surjectivity of the map $A_1(x^{k_1}) + A_2(y^{k_2}) \in \mathbf{O}(\mathbb{F})\langle x, y\rangle$ on $\mathbf{O}(\mathbb{F})$. For this, we use the orbit representatives of the ${G}_2(\mathbb{F})$-action on $\mathbf{O}(\mathbb{F}) \times \mathbf{O}(\mathbb{F}) $ for the tuple $(A_1, A_2)$, and characterize the ones which give a surjective map.

math.RA↗

arXiv2024

A survey on power maps in groups

The study of word maps on groups has been of deep interest in recent years. This survey focuses on the case of power maps on groups; $viz.$ the map $x\mapsto x^M$ for a group $G$, and an integer $M\geq 2$. Here, we accumulate various results on the subject and pose some questions.

math.GR↗

arXiv2024

Roots of identity in finite groups of Lie type

Given an integer $M\geq 2$, we deploy the generating function techniques to compute the number of $M$-th roots of identity in some of the well-known finite groups of Lie type, more precisely for finite general linear groups, symplectic groups, orthogonal groups of all types and unitary groups over finite fields of odd characteristics.

math.GR↗

arXiv2024

Fibers of the square map in some finite groups of Lie type and an application

Let $G$ be one of the finite general linear, unitary, symplectic or orthogonal groups over finite fields of odd order. We find the cardinality of the fibers of the square map at a given generic element. Using this we find the number of real conjugacy classes of $G$. This is primarily achieved by leveraging a recent solution to Brauer's problem 14.

math.GR↗

arXiv2024

Word maps, polynomial maps and image ratios

If $A$ is a finite group (or a finite ring) and $ω$ is a word map (or a polynomial map), we define the quantity $|ω(A)|/|A|$ as the image ratio of $ω$ on $A$ and will be denoted by $μ(ω,A)$. In this article, we investigate the set $\mathrm{R}(ω)=\{μ(ω,A) : A \text{ is a finite group}\}$, and also consider the case of rings. Specifically, we demonstrate the existence of word maps (and polynomial maps) whose set of image ratios is dense in $[0,1]$ for both groups (and rings).

math.GR↗

arXiv2024

Surjectivity of polynomial maps on Matrices

For $n\geq 2$, we consider the map on $M_n(\mathbb K)$ given by evaluation of a polynomial $f(X_1, \ldots, X_m)$ over the field $\mathbb K$. In this article, we explore the image of the diagonal map given by $f=δ_1 X_1^{k_1} + δ_2 X_2^{k_2} + \cdots +δ_m X_m^{k_m}$ in terms of the solution of certain equations over $\mathbb K$. In particular, we show that for $m\geq 2$, the diagonal map is surjective when (a) $\mathbb K= \mathbb C$, (b) $\mathbb K= \mathbb F_q$ for large enough $q$. Moreover, when $\mathbb K= \mathbb R$ and $m=2$ it is surjective except when $n$ is odd, $k_1, k_2$ are both even, and $δ_1δ_2>0$ (in that case the image misses negative scalars), and the map is surjective for $m\geq 3$. We further show that on $M_n(\mathbb H)$ the diagonal map is surjective for $m\geq 2$, where $\mathbb H$ is the algebra of Hamiltonian quaternions.

math.GR↗

arXiv2024

Character covering number of $\mathrm{PSL}_2 (q)$

For a group $G$ and a character $χ$ of $G$, let $c(χ)$ denote the set of all irreducible characters of $G$, occurring in $χ$. We prove that whenever $q\geq 8$, all non-trivial irreducible character $χ$ of $\mathrm{PSL}_2(q)$ satisfies $c(χ^4)=\mathrm{Irr}\left(\mathrm{PSL}_2(q)\right)$ if $q=2^{2m+1}$ and $c(χ^3)=\mathrm{Irr}\left(\mathrm{PSL}_2(q)\right)$ otherwise.

math.GR↗

arXiv2023

Images of polynomial maps with constants

Let $K$ be an algebraically closed field and $\mathrm{M}(2,K)$ be the $2\times 2$ matrix algebra over $K$ and $\mathrm{GL}(2,K)$ be the invertible elements in $\mathrm{M}(2,K)$. We explore the image of polynomials with constants, namely from the free algebra $\mathrm{M}(2,K)\langle x, y\rangle$. In this article, we compute the images of the polynomial maps given by (a) generalized sum of powers $Ax^{k_1} + By^{k_2}$ and (b) generalized commutator map $Axy -Byx$, where $A$, $B$ are non-zero elements of $\mathrm{M}(2,K)$. We compute this in the first case by fixing a simultaneous conjugate pair for $A, B$ and it turns out that it is surjective in most of the cases. In the second case, we show that the image of the map is always a vector space.

math.GR↗

arXiv2023

Hopf Galois structures, skew braces for groups of size $p^nq$: The cyclic Sylow subgroup case

Let $n\geq 1$ be an integer, $p$, $q$ be distinct odd primes. Let ${G}$, $N$ be two groups of order $p^nq$ with their Sylow-$p$-subgroups being cyclic. We enumerate the Hopf-Galois structures on a Galois ${G}$-extension, with type $N$. This also computes the number of skew braces with additive group isomorphic to $G$ and multiplicative group isomorphic to $N$. Further when $q<p$, we give a complete classification of the Hopf-Galois structures on Galois-$G$-extensions.

math.GR↗

arXiv2023

Powers in finite unitary groups

Let $\text{U}(n,\mathbb{F}_{q^2})$ denote the subgroup of unitary matrices of the general linear group $\text{GL}(n,\mathbb{F}_{q^2})$ which fixes a Hermitian form and $M\geq 2$ an integer. This is a companion paper to the previous works where the elements of the groups $\text{GL}(n,\mathbb{F}_{q})$, $\text{Sp}(2n,\mathbb{F}_{q})$, $\text{O}^{\pm}(2n,\mathbb{F}_{q})$ and $\text{O}(2n+1,\mathbb{F}_{q})$ which has an $M$-th root in the concerned group, have been described. Here we will describe the $M$-th powers in unitary groups for the regular semisimple, semisimple and cyclic elements. Our methods are parallel to those of the Memoir ``A generating function approach to the enumeration of matrices in classical groups over finite fields" by Fulman, Neumann and Praeger.

math.GR↗

arXiv2023

The image of polynomials and Waring type problems on upper triangular matrix algebras

Let $p$ be a polynomial in non-commutative variables $x_1,x_2,\ldots,x_n$ with constant term zero over an algebraically closed field $K$. The object of study in this paper is the image of this kind of polynomial over the algebra of upper triangular matrices $T_m(K)$. We introduce a family of polynomials called multi-index $p$-inductive polynomials for a given polynomial $p$. Using this family we will show that, if $p$ is a polynomial identity of $T_t(K)$ but not of $T_{t+1}(K)$, then $p \left(T_m(K)\right)\subseteq T_m(K)^{(t-1)}$. Equality is achieved in the case $t=1,~m-1$ and an example has been provided to show that equality does not hold in general. We further prove existence of $d$ such that each element of $T_m(K)^{(t-1)}$ can be written as sum of $d$ many elements of $p\left( T_m(K) \right)$. It has also been shown that the image of $T_m(K)^\times$ under a word map is Zariski dense in $T_m(K)^\times$.

math.RA↗

arXiv2023

Cycle matrices: A combinatorial approach to the set-theoretic solutions of the Quantum Yang-Baxter Equation

An $n\times n$ matrix $M=[m_{ij}]$ with $m_{ij}\in U_n=\{1,2,\ldots,n\}$ will be called a cycle matrix if $(U_n,\cdot)$ is a cycle set, where $i\cdot j=m_{ij}$. We study these matrices in this article. Using these matrices, we give some recipes to construct solutions, which include the multipermutation level $2$ solutions. As an application of these, we construct a multi-permutation solution of level $r$ for all $r\geq 1$. Our method gives alternate proof that the class of permutation groups of solutions contains all finite abelian groups.

math.GR↗