arXiv subjects

Soham Swadhin Pradhan

Publications and source records attributed to Soham Swadhin Pradhan.

8 recordsLinked to original sources

arXiv2021

On the Vanishing Criterion for the Cohomology Groups of the Automorphism Group of a finite Abelian $p$-Group

For a partition $\underlineλ = (λ_{1}^{ρ_1}>λ_{2}^{ρ_2}>λ_{3}^{ρ_3}>\ldots>λ_{k}^{ρ_k})$ and its associated finite abelian $p$-group $\mathcal{A}_{\underlineλ}=\underset{i=1}{\overset{k}{\oplus}} (\mathbb{Z}/p^{λ_i}\mathbb{Z})^{ρ_i}$, where $p$ is a prime, we consider two actions of its automorphism group $\mathcal{G}_{\underlineλ}$ on $\mathcal{A}_{\underlineλ}$. The first action is the natural action $g\bullet a=\ ^ga$ for all $g\in\mathcal{G}_{\underlineλ}$ and $a\in\mathcal{A}_{\underlineλ}$ where the action map is denoted by $Λ_1=Id_{\mathcal{G}_{\underlineλ}}:\mathcal{G}_{\underlineλ}\longrightarrow \mathcal{G}_{\underlineλ}$ and the second action is the trivial action $g\bullet a=a$ for all $g\in\mathcal{G}_{\underlineλ}$ and $a\in\mathcal{A}_{\underlineλ}$ where the action map is denoted by $Λ_2:\mathcal{G}_{\underlineλ}\longrightarrow \{e\}\subset\mathcal{G}_{\underlineλ}$ the trivial map. For the natural action $Λ_1$, we show that the first and second cohomology groups $H_{Λ_1}^i(\mathcal{G}_{\underlineλ},\mathcal{A}_{\underlineλ}),i=1,2$ vanish for any partition $\underlineλ$ for an odd prime $p$. For the trivial action $Λ_2$ we show that, for an odd prime $p$, the first cohomology group $H_{Λ_2}^1(\mathcal{G}_{\underlineλ},\mathcal{A}_{\underlineλ})$ and for an odd prime $p\neq 3$, the second cohomology group $H_{Λ_2}^2(\mathcal{G}_{\underlineλ},\mathcal{A}_{\underlineλ})$ vanish if and only if the difference between two successive parts of the partition $\underlineλ$ is at most one. This is done by using the $mod\ p$ cohomologies $H^i(\mathcal{G}_{\underlineλ},\mathbb{Z}/p\mathbb{Z}),i=1,2$.

math.GR↗

arXiv2021

Schur Index and Extensions of Witt-Berman's Theorems

Let $G$ be a finite group, and $F$ a field of characteristic $0$ or prime to the order of $G$. In $1952$, Witt and in $1956$, Berman independently proved that the number of inequivalent irreducible $F$-representations of $G$ is equal to the number of $F$-conjugacy classes of the elements of $G$, where "$F$-conjugacy" was defined in a certain way. In this paper, we define $F$-conjugacy on $G$ in a natural way and give a proof of the above Witt-Berman theorem. In addition, we give an explicit formula for computing a primitive central idempotent (pci) of the group algebra $F[G]$ corresponding to an irreducible $F$-representation of $G$, which can be obtained from the "$F$-character table" of $G$. Let $G$ be a finite group with a normal subgroup $H$ of index $p$, a prime. In $1955$, in case $F$ is algebraically closed, Berman computed the primitive central idempotent (pci) of $F[G]$ corresponding to an irreducible $F$-representation of $G$, in terms of pci's of $F[H]$. In this paper, we give a complete proof of this Berman's theorem, and extend this result when $F$ is not necessarily algebraically closed. Also, using classical Schur's theory and Wedderburn's theory, we work out decomposition of induced representation of an irreducible $F$-representation of $H$, into irreducible components.

math.RT↗

arXiv2021

Degrees of Faithful Irreducible Representations of Certain Metabelian Groups and a Question of Sim

In this paper, we answer affirmatively a question of H S Sim on representations in characteristic $0$, for a class of metabelian groups. Moreover, we provide examples to point out that the analogous answer is no longer valid if the solvable group has derived length larger than 2. Let $F$ be a field of characteristic $0$ and $\overline{F}$ be its algebraic closure. We prove that if $G$ is a finite metabelian group containing a maximal abelian normal subgroup which is a p-group with abelian quotient, all possible faithful irreducible representations over $F$ have the same degree and that the Schur index of any faithful irreducible $\overline{F}$-representation with respect to $F$ is always $1$ or $2$. H S Sim had proven such a result for metacyclic groups when the characteristic of $F$ is positive and posed the question in characteristic $0$. Our result answers this question for the above class of metabelian groups affirmatively. We also determine explicitly the Wedderburn component corresponding to any faithful irreducible $\overline{F}$-representation in the group algebra $F[G]$.

math.RT↗

arXiv2021

Rational and Quasi-Permutation Representations of Holomorph of Cyclic p-Groups

For a finite group $G$, let $p(G)$ denote the minimal degree of a faithful permutation representation of $G$. The minimal degree of a faithful representation of $G$ by quasi-permutation matrices over the fields $\mathbb{C}$ and $\mathbb{Q}$ are denoted by $c(G)$ and $q(G)$ respectively. In general $c(G)\leq q(G)\leq p(G)$ and either inequality may be strict. In this paper, we study the representation theory of the group $G =$ Hol$(C_{p^{n}})$, which is the holomorph of a cyclic group of order $p^n$, $p$ a prime. This group is metacyclic when $p$ is odd and metabelian but not metacyclic when $p=2$ and $n \geq 3$. We explicitly describe the set of all isomorphism types of irreducible representations of $G$ over the field of complex numbers $\mathbb{C}$ as well as the isomorphism types over the field of rational numbers $\mathbb{Q}$. We compute the Wedderburn decomposition of the rational group algebra of $G$. Using the descriptions of the irreducible representations of $G$ over $\mathbb{C}$ and over $\mathbb{Q}$, we show that $c(G) = q(G) = p(G) = p^n$ for any prime $p$. The proofs are often different for the case of $p$ odd and $p=2$.

math.RT↗

arXiv2019

On the Endomorphism Semigroups of Extra-special $p$-groups and Automorphism Orbits

For an odd prime $p$ and a positive integer $n$, it is well known that there are two types of extra-special $p$-groups of order $p^{2n+1}$, first one is the Heisenberg group which has exponent $p$ and the second one is of exponent $p^2$. In this article, a new way of representing the extra-special $p$-group of exponent $p^2$ is given. These representations facilitate an explicit way of finding formulae for any endomorphism and any automorphism of an extra-special $p$-group $G$ for both the types. Based on these formulae, the endomorphism semigroup $End(G)$ and the automorphism group $Aut(G)$ are described. The endomorphism semigroup image of any element in $G$ is found and the orbits under the action of the automorphism group $Aut(G)$ are determined. As a consequence it is deduced that, under the notion of degeneration of elements in $G$, the endomorphism semigroup $End(G)$ induces a partial order on the automorphism orbits when $G$ is the Heisenberg group and does not induce when $G$ is the extra-special $p$-group of exponent $p^2$. Finally we prove that the cardinality of isotropic subspaces of any fixed dimension in a non-degenerate symplectic space is a polynomial in $p$ with non-negative integer coefficients. Using this fact we compute the cardinality of $End(G)$.

math.GR↗

arXiv2018

Representations and primitive central idempotents of a finite solvable group

Let $G$ be a finite solvable group. Then $G$ always has a useful presentation, which we call a "long presentation". Using a "long presentation" of $G$, we present an inductive method of constructing the irreducible representations of $G$ over $\mathbb{C}$ and computing the primitive central idempotents of the complex group algebra $\mathbb{C}[G]$. For a finite abelian group, we present a systematic method of constructing the irreducible representations over a field of characteristic either $0$ or prime to order of the group and also a systematic method of computing the primitive central idempotents of the semisimple abelian group algebra.

math.RT↗

arXiv2018

Algorithmic construction of representations of finite solvable groups

The dominant theme of this thesis is the construction of matrix representations of finite solvable groups using a suitable system of generators. For a finite solvable group $G$ of order $N = p_{1}p_{2}\dots p_{n}$, where $p_{i}$'s are primes, there always exists a subnormal series: $\langle {e} \rangle = G_{o} < G_{1} < \dots < G_{n} = G$ such that $G_{i}/G_{i-1}$ is isomorphic to a cyclic group of order $p_{i}$, $i = 1,2,\dots,n$. Associated with this series, there exists a system of generators consisting $n$ elements $x_{1}, x_{2}, \dots, x_{n}$ (say), such that $G_{i} = \langle x_{1}, x_{2}, \dots, x_{i} \rangle$, $i = 1,2,\dots,n$, which is called a "long system of generators". In terms of this system of generators and conjugacy class sum of $x_{i}$ in $G_{i}$, $i = 1,2, \dots, n$, we present an algorithm for constructing the irreducible matrix representations of $G$ over $\mathbb{C}$ within the group algebra $\mathbb{C}[G]$. This algorithmic construction needs the knowledge of primitive central idempotents, a well defined set of primitive (not necessarily central) idempotents and the "diagonal subalgebra" of $\mathbb{C}[G]$. In terms of this system of generators, we give simple expressions for the primitive central idempotents, a well defined system of primitive (not necessarily central) idempotents and a convenient set of generators of the "diagonal subalgebra" of $\mathbb{C}[G]$. For a finite abelian group, we present an algorithm for constructing the inequivalent irreducible matrix representations over a field of characteristic $0$ or prime to the order of the group and a systematic way of computing the primitive central idempotents of the group algebra. Besides that, we give simple expressions of the primitive central idempotents of the rational group algebra of a finite abelian group using a "long presentation" and it's Wedderburn decomposition.

math.RT↗

arXiv2018

Representations induced from a normal subgroup of prime index

Let $G$ be a finite group, $H$ be a normal subgroup of prime index $p$. Let $F$ be a field of either characteristic $0$ or prime to $|G|$. Let $η$ be an irreducible $F$-representation of $H$. If $F$ is an algebraically closed field of characteristic either $0$ or prime to $|G|$, then the induced representation $η\uparrow^{G}_{H}$ is either irreducible or a direct sum of $p$ pairwise inequivalent irreducible representations. In this paper, we show that if $F$ is not assumed algebraically closed field, then there are five possibilities in the decomposition of induced representation into irreducible representations.

math.RT↗