arXiv subjects
Publications and source records attributed to Anupam Singh.
This paper presents some algorithms in linear algebraic groups. These algorithms solve the word problem and compute the spinor norm for orthogonal groups. This gives us an algorithmic definition of the spinor norm. We compute the double coset decomposition with respect to a Siegel maximal parabolic subgroup, which is important in computing infinite-dimensional representations for some algebraic groups.
Let $G$ be a finite group. This expository article explores the subject of commuting probability in the group $G$ and its relation with simultaneous conjugacy classes of commuting tuples in $G$. We also point out the relevance of this topic in context of topology where similar problems are studied when $G$ is a Lie group.
This paper concerns the enumeration of simultaneous conjugacy classes of tuples of commuting unitary matrices and of commuting symplectic matrices over a finite field $\mathbf{F}_q$ of odd size. For any given conjugacy class, the orbits for the action of its centralizer group on itself by conjugation are called branches. We determine the branching rules for the unitary groups $U_2(\mathbf{F}_q), U_3(\mathbf{F}_q)$, and for the symplectic groups $Sp_2(\mathbf{F}_q), Sp_4(\mathbf{F}_q)$.
Dark Energy is the dominant component of the energy density of the Universe. However, it is also very elusive since its interaction with the rest of the Universe is primarily gravitational.Since Dark Energy is a low energy phenomenon from the perspective of particle physics and field theory, a fundamental approach based on fields in curved space is sufficient to understand the current dynamics of Dark Energy. The key issue is to understand the gravitational dynamics of Dark Energy and its observational consequences. However, finding the observational consequences of Dark Energy dynamics has been a very challenging task. For something which is the dominant component of the energy density of the Universe, Dark Energy appears to be very distant and reclusive. Here we show that the Dark Energy dynamics results in the production of gravitational waves which produce the ellipticity variation in the orbit of earth that results in the periodicity of the Ice Ages observed and documented by geologists and climatologists. Previously, no observational signature of gravitational waves produced by Dark Energy dynamics has been reported. Further, no interpretation of the ellipticity variation of the orbit of earth due to gravitational waves or the linking of such gravitational waves to the Ice Age periodicity has been reported previously. We hope that the current work will lead to some fresh insights and some more interesting work.
Dark Energy is the dominant component of the energy density of the universe. In a previous paper, we have shown that the collapse of dark energy fields leads to the formation of Super Massive Black Holes with masses comparable to the masses of Black Holes at the centers of galaxies. Thus it becomes a pressing issue to investigate the other physical consequences of the collapse of Dark Energy fields. Given that the primary interactions of Dark Energy fields with the rest of the Universe are gravitational, it is particularly interesting to investigate the gravitational wave signals emitted during the process of the collapse of Dark Energy fields. This is the focus of the current work described in this paper. We describe and use the 3+1 BSSN formalism to follow the evolution of the dark energy fields coupled with gravity and to extract the gravitational wave signals. Finally, we describe the results of our numerical computations and the gravitational wave signals produced as a result of the collapse of the dark energy fields.
Let $k$ be a perfect field such that for every $n$ there are only finitely many field extensions, up to isomorphism, of $k$ of degree $n$. If $G$ is a reductive algebraic group defined over $k$, whose characteristic is very good for $G$, then we prove that $G(k)$ has only finitely many $z$-classes. For each perfect field $k$ which does not have the above finiteness property we show that there exist groups $G$ over $k$ such that $G(k)$ has infinitely many $z$-classes.
Let $k$ be a field of characteristic $\neq 2$. In this paper we study squares, cubes and their products in split and anisotropic groups of type $A_1$. In split case, we show that computing $n^{\rm th}$ roots is equivalent to finding solutions of certain polynomial equations in at most two variables over the base field $k$. The description of these polynomials involves generalised Fibonacci polynomials. Using this we obtain asymptotic proportions of $n^{\rm th}$ powers, and conjugacy classes which are $n^{\rm th}$ powers, in ${\rm SL}_2(\mathbb F_q)$ when $n$ is a prime or $n = 4$. We also extend already known Waring type result for ${\rm SL}_2(\mathbb F_q)$, that every element of ${\rm SL}_2(\mathbb F_q)$ is a product of two squares, to ${\rm SL}_2(k)$ for an arbitrary $k$. For anisotropic groups of type $A_1$, namely ${\rm SL}_1(Q)$ where $Q$ is a quaternion division algebra, we prove that when $2$ is a square in $k$, every element of ${\rm SL}_1(Q)$ is a product of two squares if and only if $-1$ is a square in ${\rm SL}_1(Q)$.
This expository article revolves around the question to find short presentations of finite simple groups. This subject is one of the most active research areas of group theory in recent times. We bring together several known results on two-generation and $(2,3)$-generation of finite simple groups and how it impacts computational group theory.
Let $G$ be a group. Two elements $x,y \in G$ are said to be in the same $z$-class if their centralizers in $G$ are conjugate within $G$. Consider $\mathbb F$ a perfect field of characteristic $\neq 2$, which has a non-trivial Galois automorphism of order $2$. Further, suppose that the fixed field $\mathbb F_0$ has the property that it has only finitely many field extensions of any finite degree. In this paper, we prove that the number of $z$-classes in the unitary group over such fields is finite. Further, we count the number of $z$-classes in the finite unitary group $U_n(q)$, and prove that this number is same as that of $GL_n(q)$ when $q>n$.
This is an exercise based approach to matrix groups. The idea is to collect a bunch of exercises at one place which anyone with basic knowledge of linear algebra can attempt to solve and learn matrix groups and algebraic groups.
Let $q$ be a $p$-power where $p$ is a fixed prime. In this paper, we look at the $p$-power maps on unitriangular group $U(n,q)$ and triangular group $T(n,q)$. In the spirit of Borel dominance theorem for algebraic groups, we show that the image of this map contains large size conjugacy classes. For the triangular group we give a recursive formula to count the image size.
In the paper we consider images of finite simple projective special linear and unitary groups under power words. In particular, we show that if $G\simeq \PSL_n^\varepsilon (q)$, then for every power words of type $x^M$ there exist constant $c$ and $N$ such that $\vert ω(G)\vert >c\frac{\vert G\vert }{n}$ whenether $\vert G\vert >N$.
This paper studies an algorithm similar to that of Gaussian elimination in symplectic and orthogonal groups. We discuss two applications of this algorithm in computational group theory. One computes the spinor norm and the other computes the double coset decomposition with respect to Siegel maximal parabolic subgroup.
In this paper, we compute the number of z-classes (conjugacy classes of centralizers of elements) in the symmetric group S_n, when n is greater or equal to 3 and alternating group A_n, when n is greater or equal to 4. It turns out that the difference between the number of conjugacy classes and the number of z-classes for S_n is determined by those restricted partitions of n-2 in which 1 and 2 do not appear as its part. And, in the case of alternating groups, it is determined by those restricted partitions of n-3 which has all its parts distinct, odd and in which 1 (and 2) does not appear as its part, along with an error term. The error term is given by those partitions of n which have each of its part distinct, odd and perfect square. Further, we prove that the number of rational-valued irreducible complex characters for A_n is same as the number of conjugacy classes which are rational.
Gaussian elimination is used in special linear groups to solve the word problem. In this paper, we extend Gaussian elimination to unitary groups. These algorithms have an application in building a public-key cryptosystem, we demonstrate that.
In this paper we study the MOR cryptosystem using finite classical Chevalley groups over a finite field of odd characteristic. In the process we develop an algorithm for these Chevalley groups in the same spirit as the row-column operation for special linear group. We focus our study on orthogonal and symplectic groups. We find the hardness of the proposed MOR cryptosystem for these groups.
We classify all strongly real conjugacy classes of the finite unitary group $\U(n, F_q)$ when $q$ is odd. In particular, we show that $g \in \U(n, F_q)$ is strongly real if and only if $g$ is an element of some embedded orthogonal group $O^{\pm}(n, F_q)$. Equivalently, $g$ is strongly real in $\U(n, F_q)$ if and only if $g$ is real and every elementary divisor of $g$ of the form $(t \pm 1)^{2m}$ has even multiplicity. We apply this to obtain partial results on strongly real classes in the finite symplectic group $\Sp(2n, F_q)$, $q$ odd, and a generating function for the number of strongly real classes in $\U(n, F_q)$, $q$ odd, and we also give partial results on strongly real classes in $\U(n, F_q)$ when $q$ is even.
In this article we try to explore the relation between real conjugacy classes and real characters of finite groups at more refined level. This refinement is in terms of properties of groups such as strong reality and total orthogonality. In this connection we raise several questions and record several examples which have motivated those questions.