Full Symmetric Group Research Articles

Galois groups of n 0 + n 1 X + ⋯ + n 6 X6

We show that the Galois group of the polynomial in the title is isomorphic to the full symmetric group on six symbols for all but finitely many [Formula: see text]. This complements earlier work of Filaseta and Moy, who studied Galois groups of [Formula: see text] for more general pairs [Formula: see text], but had to admit a possibly infinite exceptional set specifically for [Formula: see text] of at most logarithmic growth in [Formula: see text]. The proof rests upon invoking Faltings’ theorem on a suitable fibration of Galois resolvents to show that this exceptional set is, in fact, finite.

The Galois group of X^{p^2}+aX+a

Let p be an odd prime number, and a be an integer divisible by p exactly once. We prove that the Galois group G of the trinomial X^{p^{2}}+aX+a over the field Q of rational number is either the full symmetric group S_{p^{2}} or G lies between AGL(1,p^{2}) and AGL(2,p)$. Furthermore, we establish conditions when G is S_{p^{2}}.

GALOIS GROUP OF Xpn+aX+a

Let p be an odd prime number, n≥3 be a rational integer, and let f(X)=Xpn+aX+a∈ℤ[X] be an Eisenstein trinomial with respect to p. We prove that the Galois group G of f(X) over the field ℚ of rational numbers, is either the full symmetric group Spn, or AGL(1,pn)≤G≤AGL(n,p). We also show that G≃Spn, except possibly when |pnpn−1+ap(pn−1)pn−1| is a square, and for each prime divisor ℓ of a∕p, p divides the ℓ-adic valuation vℓ(a) of the integer a.

Distinguishing Number of Universal Homogeneous Urysohn Metric Spaces

The distinguishing number of a structure is the smallest size of a partition of its elements so that only the trivial automorphism of the structure preserves the partition. We show that for any countable subset of the positive real numbers, the corresponding countable Urysohn metric space, when it exists, has distinguishing number two or is infinite.
 While it is known that a sufficiently large finite primitive structure has distinguishing number two, unless its automorphism group is not the full symmetric group or alternating group, the infinite case is open and these countable Urysohn metric spaces provide further confirmation toward the conjecture that all primitive homogeneous countably infinite structures have distinguishing number two or else is infinite.

On quotients ofℳ¯g,nby certain subgroups ofSn

We investigate when certain quotients of the compactified moduli space of n-pointed genus g curves ℳ ¯ G :=ℳ ¯ g,n /G are of general type or, on the contrary, uniruled, for a fairly broad class of subgroups G of the symmetric group S n which act by permuting the n marked points. We show that the property of being of general type only depends on the transpositions contained in G. Furthermore, in the case that G is the full symmetric group S n or a product S n 1 ×⋯×S n m , we find a narrow transitional band in which ℳ ¯ G changes its behaviour from being of general type to its opposite, i.e. being uniruled, as n increases. As an application we consider the universal difference variety ℳ ¯ g,2n /S n ×S n .

Monodromy of rational curves on K3 surfaces of low genus

In many situations, the monodromy group of enumerative problems will be the full symmetric group. In this paper, we study a similar phenomenon on the rational curves in $|\mathcal{O}(1)|$ on a generic K3 surface of fixed genus over $\mathbb{C}$ as the K3 surface varies. We prove that when the K3 surface has genus $g$, $1\leq g\leq 3$, the monodromy group is the full symmetric group.

Searching for rigidity in algebraic starscapes

We create plots of algebraic integers in the complex plane, exploring the effect of sizing the points according to various arithmetic invariants. We focus on Galois theoretic invariants, in particular creating plots which emphasize algebraic integers whose Galois group is not the full symmetric group−these integers we call rigid. We then give some analysis of the resulting images, suggesting avenues for future research about the geometry of so-called rigid algebraic integers.

Sectional monodromy groups of projective curves

Fix a degree $d$ projective curve $X \subset \mathbb{P}^r$ over an algebraically closed field $K$. Let $U \subset (\mathbb{P}^r)^*$ be a dense open subvariety such that every hyperplane $H \in U$ intersects $X$ in $d$ smooth points. Varying $H \in U$ produces the monodromy action $\varphi: \pi_1^{\text{\'et}}(U) \to S_d$. Let $G_X := \mathrm{im}(\varphi)$. The permutation group $G_X$ is called the sectional monodromy group of $X$. In characteristic zero $G_X$ is always the full symmetric group, but sectional monodromy groups in characteristic $p$ can be smaller. For a large class of space curves ($r \geqslant 3$) we classify all possibilities for the sectional monodromy group $G$ as well as the curves with $G_X=G$. We apply similar methods to study a particular family of rational curves in $\mathbb{P}^2$, which enables us to answer an old question about Galois groups of generic trinomials.

English

Let V be a unitary space. For an arbitrary subgroup G of the full symmetric group Sm and an arbitrary irreducible unitary representation Λ of G, we study the generalized symmetry class of tensors over V associated with G and Λ. Some important properties of this vector space are investigated.

WILD RAMIFICATION IN TRINOMIAL EXTENSIONS AND GALOIS GROUPS

AbstractIt is proven that, for a wide range of integers s (2 < s < p − 2), the existence of a single wildly ramified odd prime l ≠ p leads to either the alternating group or the full symmetric group as Galois group of any irreducible trinomial Xp + aXs + b of prime degree p.

A Nonsingular Action of the Full Symmetric Group Admits an Equivalent Invariant Measure

A Nonsingular Action of the Full Symmetric Group Admits an Equivalent Invariant Measure

Galois groups of Taylor polynomials of some elementary functions

Motivated by Schur’s result on computing the Galois groups of the exponential Taylor polynomials, this paper aims to compute the Galois groups of the Taylor polynomials of the elementary functions [Formula: see text] and [Formula: see text]. We first show that the Galois groups of the [Formula: see text]th Taylor polynomials of [Formula: see text] are as large as possible, namely, [Formula: see text] (full symmetric group) or [Formula: see text] (alternating group), depending on the residue of the integer number [Formula: see text] modulo [Formula: see text]. We then compute the Galois groups of the [Formula: see text]th Taylor polynomials of [Formula: see text] and show that these Galois groups essentially coincide with the Coexter groups of type [Formula: see text] (or an index 2 subgroup of the corresponding Coexter group).

Quantum-mechanical definition of atoms and their mutual interactions in Born-Oppenheimer molecules

The apparent absence of meaningful assignments of electrons and indistinguishable nuclei to particular atoms in a chemical aggregate would seem to preclude the quantum-mechanical definition of atomic Hamiltonian operators within molecules and matter. The electronic energies of individual constituent atoms, as well as the interactions between them, are accordingly widely perceived as objectively undefined in molecular quantum theory, requiring additional auxiliary conditions to achieve quantitative specificity, giving rise to a plethora of individual preferences. Here we address the issue of assignments of electrons to atoms within molecules at the Born-Oppenheimer classical fixed-nuclei level of theory and provide thereby quantum-mechanical definitions of atomic operators and of the interactions between them. In the spirit of early work of Longuet-Higgins, a van der Waals subgroup of the full molecular electronic symmetric group is shown to facilitate assignments of electrons to particular atomic nuclei in a molecule. The orthonormal (Eisenschitz-London) outer products of atomic eigenstates that provide separable Hilbert space representations of this symmetric subgroup furthermore support totally antisymmetric solutions of the molecular Schr\odinger equation. Self-adjoint atomic and atomic-interaction operators within a molecule defined in this way are seen to have universal Hermitian matrix representatives and physically significant expectation values in totally antisymmetric molecular eigenstates. Adiabatic Born-Oppenheimer molecular energies emerge naturally from the development in the form of sums of the energies of individual atomic constituents and of their atomic-pairwise interactions, in the absence of additional auxiliary conditions. A detailed and nuanced quantitative description of electronic structure and bonding is provided thereby which includes the interplay between atomic promotion and interaction energies, common representations of atomic-state hybridization and interatomic charge apportionment, potentially measurable multiatom entanglements upon coherent dissociations of molecules, and other attributes of the development revealed by selected illustrative calculations. These include applications to the ground and electronically excited states of diatomic and triatomic hydrogen molecules, which exhibit significant accommodation among the atomic promotion and interaction energies, as well as entanglements among atomic states, over the entire range of molecular geometries traversed in the course of two- and three-atom dissociations.

Mahler measure on Galois extensions

We study the Mahler measure of generators of a Galois extension with Galois group the full symmetric group. We prove that two classical constructions of generators give always algebraic numbers of big height. These results answer a question of Smyth and provide some evidence to a conjecture which asserts that the height of such a generator grows to infinity with the degree of the extension.

On Minimal Generating Sets for Symmetric and Alternating Groups

By a famous result, the subgroup generated by the n-cycle $$\sigma =(1,2,\ldots ,n)$$ and the transposition $$\tau =(a,b)$$ is the full symmetric group $$S_{n}$$ if and only if $$gcd(n,b-a)=1$$ . In this paper, we first generalize the above result for one n-cycle and k arbitrary transpositions, and then provide similar necessary and sufficient conditions for the subgroups of $$S_{n}$$ in the following three cases: first, the subgroup generated by the n-cycle $$\sigma $$ and a 3-cycle $$\delta =(a,b,c)$$ , second, the subgroup generated by the n-cycle $$\sigma $$ and a set of transpositions and 3-cycles, and third, by the n-cycle $$\sigma $$ and an involution (a, b)(c, d). In the first case, we also determine the structure of the subgroup generated by $$(1,2,\ldots ,n)$$ and a 3-cycle $$\delta =(a,b,c)$$ in general. Finally, an application to unsolvability of a certain infinite family of polynomials by radicals is given.

Numerical Computation of Galois Groups

The Galois/monodromy group of a family of geometric problems or equations is a subtle invariant that encodes the structure of the solutions. We give numerical methods to compute the Galois group and study it when it is not the full symmetric group. One algorithm computes generators, while the other studies its structure as a permutation group. We illustrate these algorithms with examples using a Macaulay2 package we are developing that relies upon Bertini to perform monodromy computations.

Two-dimensional families of hyperelliptic Jacobians with big monodromy

Let $K$ be a global field of characteristic different from 2 and $u(x)\in K[x]$ be an irreducible polynomial of even degree $2g\ge 6$, whose Galois group over $K$ is either the full symmetric group $S_{2g}$ or the alternating group $A_{2g}$. We describe explicitly how to choose (infinitely many) pairs of distinct elements $t_1, t_2$ of $K$ such that the $g$-dimensional jacobian of a hyperelliptic curve $y^2=(x-t_1)(x-t_2))u(x)$ has no nontrivial endomorphisms over an algebraic closure of $K$ and has big $\ell$-adic monodromy.

On Derivatives and Norms of Generalized Matrix Functions and Respective Symmetric Powers

In recent papers, S. Carvalho and P. J. Freitas obtained formulas for directional derivatives, of all orders, of the immanant and of the m-th $\xi$-symmetric tensor power of an operator and a matrix, when $\xi$ is a character of the full symmetric group. The operator bound norm of these derivatives was also calculated. In this paper similar results are established for generalized matrix functions and for every symmetric tensor power.

On permutation modules and decomposition numbers of the symmetric group

We study the indecomposable summands of the Foulkes permutation module obtained by inducing the trivial F(Sa≀Sn)-module to the full symmetric group San for any field F of odd prime characteristic p such that a<p≤n. In particular, we characterize the vertices of such indecomposable summands. As a corollary we disprove a modular version of Foulkes' Conjecture.In the second part of the article we use this information to give a new description of some columns of the decomposition matrices of symmetric groups in terms of the ordinary character ϕ(an) of the Foulkes permutation module.

On the Symmetry Classes of Tensors Associated with Certain Frobenius Groups

In this paper, we study the symmetry classes of tensors associated with some Frobenius groups of order pq, where q|p-1as a subgroups of the full symmetric group on p letters. We calculate the dimension of the symmetry classes of tensor associated with some Frobenius groups and some irreducible complex characters and we obtain two useful corollary with an example.