Automorphism Of Order Research Articles

Automorphisms of quartic surfaces and Cremona transformations

In this paper, we consider the problem of determining which automorphisms of a smooth quartic surface S⊂P3 are induced by a Cremona transformation of P3. We provide the first steps towards a complete solution of this problem when ρ(S)=2. In particular, we give several examples of quartics whose automorphism groups are generated by involutions, but no non-trivial automorphism is induced by a Cremona transformation of P3, giving a negative answer for Oguiso's question of whether every automorphism of finite order of a smooth quartic surface S⊂P3 is induced by a Cremona transformation.

On the degrees of irreducible characters fixed by some field automorphism in finite groups

AbstractWe prove a variant of the theorem of Ito–Michler, investigating the properties of finite groups where a prime number does not divide the degree of any irreducible character left invariant by some Galois automorphism of order .

Inequivalent Hadamard Matrices of Order 40 using Block Designs

This article presents the classification of Hadamard matrices of order 40 with automorphisms of order 19. We determine the 2-(19,9,8) and 2-(20,10,9) designs with automorphisms of order 19. These designs are embedded into 2-(39,19,9) designs which are then extended to 3-(40,20,9) designs. We get 33 non-isomorphic 2-(39,19,9) designs out of which 18 designs gives non-isomorphic 3-(40,20,9) designs and hence 18 inequivalent Hadamard matrices of order 40 with automorphisms of order 19.

Polarized K3 surfaces with an automorphism of order 3 and low Picard number

Polarized K3 surfaces with an automorphism of order 3 and low Picard number

On parallelisms of PG(3,4) with automorphisms of order 2

Let PG(n,q) be the n-dimensional projective space over the finite field Fq. A spread in PG(n,q) is a set of mutually skew lines which partition the point set. A parallelism is a partition of the set of lines by spreads. The classification of parallelisms in small finite projective spaces is of interest for problems from projective geometry, design theory, network coding, error-correcting codes, and cryptography. All parallelisms of PG(3,2) and PG(3,3) are known. Parallelisms of PG(3,4) which are invariant under automorphisms of odd prime orders have also been classified. The present paper contributes to the classification of parallelisms in PG(3,4) with automorphisms of even order. We focus on cyclic groups of order four and the group of order two generated by a Baer involution. We examine invariants of the parallelisms such as the full automorphism group, the type of spreads and questions of duality. The results given in this paper show that the number of parallelisms of PG(3,4) is at least 8675365. Some future directions of research are outlined.

The irreducible weak modules for the fixed point subalgebra of the vertex algebra associated to a non-degenerate even lattice by an automorphism of order 2 (Part 2)

The irreducible weak modules for the fixed point subalgebra of the vertex algebra associated to a non-degenerate even lattice by an automorphism of order 2 (Part 2)

Symmetric 2‐(36,15,6) $(36,15,6)$ designs with an automorphism of order two

AbstractBouyukliev, Fack and Winne classified all 2‐ designs that admit an automorphism of odd prime order, and gave a partial classification of such designs that admit an automorphism of order 2. In this paper, we give the classification of all symmetric 2‐ designs that admit an automorphism of order two. It is shown that there are exactly nonisomorphic such designs, of which are self‐dual designs. The ternary linear codes spanned by the incidence matrices of these designs are computed. Among these codes, there are near‐extremal self‐dual codes with previously unknown weight distributions.

Symplectic rigidity of O’Grady’s tenfolds

We prove that any symplectic automorphism of finite order of an irreducible holomorphic symplectic manifold of O’Grady’s 10 10 -dimensional deformation type is trivial.

K3 surfaces with a symplectic automorphism of order 4

AbstractGiven , a K3 surface admitting a symplectic automorphism of order 4, we describe the isometry on . Having called and , respectively, the minimal resolutions of the quotient surfaces and , we also describe the maps induced in cohomology by the rational quotient maps and : With this knowledge, we are able to give a lattice‐theoretic characterization of , and find the relation between the Néron–Severi lattices of and in the projective case. We also produce three different projective models for and , each associated to a different polarization of degree 4 on .

Hadamard Matrices of Orders 60 and 64 with Automorphisms of Orders 29 and 31

A classification of Hadamard matrices of order $2p+2$ with an automorphism of order $p$ is given for $p=29$ and $31$. The ternary self-dual codes spanned by the newly found Hadamard matrices of order 60 with an automorphism of order 29 are computed, as well as the binary doubly even self-dual codes of length 120 with generator matrices defined by related Hadamard designs. Several new ternary near-extremal self-dual codes, as well as binary near-extremal doubly even self-dual codes with previously unknown weight enumerators are found.

Fusion rules for the fixed point subalgebra of the vertex algebra associated with a non-degenerate and non-positive definite even lattice by an automorphism of order 2

Let [Formula: see text] be the vertex algebra associated with a non-degenerate and non-positive definite even lattice [Formula: see text] and [Formula: see text] the fixed point subalgebra of [Formula: see text] under the action of the automorphism induced from the [Formula: see text]-isometry of [Formula: see text]. We determine the fusion rules for weak [Formula: see text]-modules. As an application of this result, we classify the irreducible weak modules for the fixed point subalgebra of [Formula: see text] under the action of the automorphism induced from an isometry of [Formula: see text] of order [Formula: see text]. We also show that every weak module for the same fixed point subalgebra is completely reducible.

Cyclic coverings of genus curves of Sophie Germain type

Abstract We consider cyclic unramified coverings of degree d of irreducible complex smooth genus $2$ curves and their corresponding Prym varieties. They provide natural examples of polarized abelian varieties with automorphisms of order d. The rich geometry of the associated Prym map has been studied in several papers, and the cases $d=2, 3, 5, 7$ are quite well understood. Nevertheless, very little is known for higher values of d. In this paper, we investigate whether the covering can be reconstructed from its Prym variety, that is, whether the generic Prym Torelli theorem holds for these coverings. We prove this is so for the so-called Sophie Germain prime numbers, that is, for $d\ge 11$ prime such that $\frac {d-1}2$ is also prime. We use results of arithmetic nature on $GL_2$ -type abelian varieties combined with theta-duality techniques.

Кольца инцидентности и их автоморфизмы

The paper is devoted to automorphisms of incidence algebras. Let I X R ( , ) be an incidence algebra of a preordered set X over a T-algebra R (T is a commutative ring). The algebra I X R ( , ) is assumed to satisfy a condition of sufficiently general character. It is called condition (II). In the case when the algebra I X R ( , ) satisfies condition (II), it is proved that any automorphism of such algebra after conjugation by an inner automorphism has a diagonalized form in a certain sense (Theorem 3.1). The other two main results of the paper are Theorem 4.1 and Corollary 4.2. In these propositions, in addition to condition (II), the algebra I X R ( , ) satisfies two other certain conditions. All three conditions are fulfilled if, for example, R is a local ring or a domain of principal left (right) ideals. Under these assumptions, it is proved that every automorphism of the algebra I X R ( , ) can be written as a product of inner, multiplicative, ring, and order automorphisms. These four kinds of automorphisms can be called standard. Here we consider an automorphism to be standard, if its structure is quite clear.

Note on Hamiltonian Graphs in Abelian 2-Groups

We analyze a graph G whose vertices are subgroups of ℤ2k isomorphic to ℤ2 × ℤ2. Two vertices are joined if their respective subgroups have nontrivial intersection. We prove that such a graph is 6(2k−2 − 1)-regular. If a graph is regular, a classical theorem by Ore claims that a graph is Hamiltonian if the degree of any vertex is at least one half of the number of vertices. Using Ore’s theorem, we show that G is Hamiltonian for k ∈{3, 4}. Ore’s theorem cannot be applied when k ≥ 5. Nevertheless, we manage to construct a Hamiltonian cycle for k = 5. Our construction uses orbits of one ℤ24 group under an action of an automorphism of order 31. It is highly likely that this approach could be generalized for k > 5.

Quasi-symmetric \(2\)-\((28,12,11)\) designs with an automorphism of order \(5\)

A design is called quasi-symmetric if it has only two block intersection numbers. Using a method based on orbit matrices, we classify quasi-symmetric \(2\)-\((28,12,11)\) designs with intersection numbers \(4\), \(6\), and an automorphism of order \(5\). There are exactly \(31\,696\) such designs up to isomorphism.

On symmetric \(2\)-\((70,24,8)\) designs with an automorphism of order \(6\)

In this paper we analyze possible actions of an automorphism of order six on a \(2\)-\((70, 24, 8)\) design, and give a complete classification for the action of the cyclic group of order six \(G= \langle \rho \rangle \cong Z_6 \cong Z_2 \times Z_3\), where \(\rho^3\) fixes exactly \(14\) points (blocks) and \(\rho^2\) fixes \(4\) points (blocks). Up to isomorphism there are \(3718\) such designs. This result significantly increases the number of previously known \(2\)-\((70,24,8)\) designs.

On fake ES-irreducible components of certain strata of smooth plane sextics

In this paper, we construct the first examples of what we call fake ES-irreducible components; Definition 2.8. In our way to do so, we classify the automorphism groups of smooth plane sextics that only have automorphisms of order [Formula: see text]; Theorems 2.1, 2.4 and 2.5, Corollaries 2.9 and 2.11.

Three generations of colored fermions with S_3 family symmetry from Cayley–Dickson sedenions

An algebraic representation of three generations of fermions with SU(3)_C color symmetry based on the Cayley–Dickson algebra of sedenions {mathbb {S}} is constructed. Recent constructions based on division algebras convincingly describe a single generation of leptons and quarks with Standard Model gauge symmetries. Nonetheless, an algebraic origin for the existence of exactly three generations has proven difficult to substantiate. We motivate mathbb {S} as a natural algebraic candidate to describe three generations with SU(3)_C gauge symmetry. We initially represent one generation of leptons and quarks in terms of two minimal left ideals of mathbb {C}ell (6), generated from a subset of all left actions of the complex sedenions on themselves. Subsequently we employ the S_3 automorphism of order three, which is an automorphism of mathbb {S} but not of mathbb {O}, to generate two additional generations. Given the relative obscurity of sedenions, efforts have been made to present the material in a self-contained manner.

Non-purely non-symplectic automorphisms of odd order on $K3$ surfaces

Non-purely non-symplectic automorphisms of odd order on $K3$ surfaces

A worldsheet approach to \U0001d4a9 = 1 heterotic flux backgrounds

Heterotic backgrounds with torsion preserving minimal supersymmetry in four dimensions can be obtained as orbifolds of principal T2 bundles over K3. We consider a worldsheet description of these backgrounds as gauged linear sigma-models (GLSMs) with (0, 2) supersymmetry. Such a formulation provides a useful framework in order to address the resolution of singularities of the orbifold geometries. We investigate the constraints imposed by discrete symmetries on the corresponding torsional GLSMs. In particular, the principal T2 connection over K3 is inherited from (0, 2) vector multiplets. As these vectors gauge global scaling symmetries of products of projective spaces, the corresponding K3 geometry is naturally realized as an algebraic hypersurface in such a product (or as a branched cover of it). We outline the general construction for describing such orbifolds. We give explicit constructions for automorphisms of order two and three.