Group Of Linear Automorphisms Research Articles

Tropical matrix groups

We study the structure of groups of finitary tropical matrices under multiplication. We show that the maximal groups of $$n \times n$$ tropical matrices are precisely the groups of the form $$G \times \mathbb {R}$$ where G is a group admitting a 2-closed permutation representation on n points. Each such maximal group is also naturally isomorphic to the full linear automorphism group of a related tropical polytope. Our results have numerous corollaries, including the fact that every automorphism of a projective (as a module) tropical polytope of full rank extends to an automorphism of the containing space.

Group actions on the tropical Hesse pencil

We show that the addition of points on the tropical Hesse curve can be realized via the intersection with a tropical line. Then the addition formula for the tropical Hesse curve is reduced from those for the level-three theta functions through the ultradiscretization procedure. A tropical analogue of the Hessian group, the group of linear automorphisms acting on the Hesse pencil, is also investigated; it is shown that the dihedral group of degree three is the group of linear automorphisms acting on the tropical Hesse pencil.

Automorphism subgroups of the free lie algebra of rank 3

Let Fr be a free Lie algebra over the field ℂ. We give an example of some subgroup of Aut Fr that is isomorphic to a subgroup of ℂ* but not conjugated with a subgroup of the linear automorphism group. Some questions are formulated.

The freeness conjecture for Hecke algebras of complex reflection groups, and the case of the Hessian group [formula omitted

We review the state-of-the-art concerning the freeness conjecture stated in the 1990s by Broué, Malle and Rouquier for generic Hecke algebras associated to complex reflection groups, and in particular we expose in detail one of the main differences with the ordinary case, namely the lack of 0-Hecke algebras. We end the paper by proving a new case of this conjecture, the exceptional group called G26 in the Shephard–Todd classification, namely the largest linear group of automorphisms of the Hessian configuration.

Nonabelian 1-cohomologies and conjugacy of finite subgroups in some extensions

We describe the conjugacy classes of finite subgroups in some split extensions using the notion of 1-cocycle and 1-coboundary with values in a noncommutative group. We prove that each finite subgroup in the automorphism group of a free Lie algebra of rank 3 is conjugated with a subgroup of the linear automorphism group provided that the group order does not divide the characteristic of the ground field.

Finite-dimensional vertex algebra modules over fixed point differential subfields

Let K be a differential field over C with derivation D, G a finite linear automorphism group over K which preserves D, and KG the fixed point subfield of K under the action of G. We show that every finite-dimensional vertex algebra KG-module is contained in some twisted vertex algebra K-module.

Explicit computations on the desingularized Kummer surface

We find formulas for the birational maps from a Kummer surface K and its dual K^* to their common minimal desingularization S. We show how the nodes of K blow up. Then we give a description of the group of linear automorphisms of S.

Finite-dimensional vertex algebra modules over fixed point commutative subalgebras

Let A be a connected commutative C -algebra with derivation D, G a finite linear automorphism group of A which preserves D, and R = A G the fixed point subalgebra of A under the action of G. We show that if A is generated by a single element as an R-algebra and is a Galois extension over R in the sense of M. Auslander and O. Goldman, then every finite-dimensional indecomposable vertex algebra R-module has a structure of twisted vertex algebra A-module.

Todd's maximum-volume ellipsoid problem on symmetric cones

Let V be a Euclidean Jordan algebra and let Ω be the associated symmetric cone, a self-dual homogeneous open convex cone, which is a symmetric space of noncompact type under G(Ω) (the linear automorphism group)-invariant Riemannian metric. We show that the radius of the largest ball centered at a ∈ Ω inscribed in Ω coincides with its minimum eigenvalue and then provide a proof of the problem of finding a point x ∈ Ω to maximize the product of the radii of the largest balls centered at a, b ∈ Ω and inscribed in Ω of the tangent space Tx (Ω) and its dual space Tx–1 (Ω), respectively. We obtain an explicit formula for the maxima; it is precisely the minimal eigenvalue of P (a1/2)b where P denotes the quadratic representation of V. This provides an affirmative answer to a question of Todd on the maxima.

Supplement to “The group E6(q) and graphs with a locally linear group of automorphisms” by V. I. Trofimov and R. M. Weiss

AbstractLet q be a prime power and let G be a group acting faithfully and vertex transitively on a graph such that for each vertex x, the stabilizer Gx is finite and contains a normal subgroup inducing on the set of neighbours of x a permutation group isomorphic to the linear group L5(q) acting on the 2-dimensional subspaces of a 5-dimensional vector space over Fq. In a companion paper, it is shown, except in some special situations where q = 2, that the kernel of the action of a vertex stabilizer Gx on the ball of radius 3 around x is trivial. In this paper we show that these special situations cannot occur.

The group E6(q) and graphs with a locally linear group of automorphisms

AbstractLet q be a prime power and let G be a group acting faithfully and vertex transitively on a graph such that for each vertex x, the stabilizer Gx is finite and contains a normal subgroup inducing on the set of neighbours of x a permutation group isomorphic to the linear group L5(q) acting on the 2-dimensional subspaces of a 5-dimensional vector space over Fq. It is shown, except in some special situations where q = 2, that the kernel of the action of a vertex stabilizer Gx on the ball of radius 3 around x is trivial. (These special situations are eliminated in a companion paper by the first author.) An example coming from the exceptional group E6(q) shows that the kernel of the action of Gx on the ball of radius 2 around x can be non-trivial.

The automorphism groups of linear codes and canonical representatives of their semilinear isometry classes

The main aim of the classification of linear codes is the evaluation of complete lists of representatives of the isometry classes. These classes are mostly defined with respect to linear isometry, but it is well known that there is also the more general definition of semilinear isometry taking the field automorphisms into account. This notion leads to bigger classes so the data becomes smaller. Hence we describe an algorithm that gives canonical representatives of these bigger classes by calculating a unique generator matrix to a given linear code, in a well defined manner.   &nbsp The algorithm is based on the partitioning and refinement idea which is also used to calculate the canonical labeling of a graph [12] and it similarly returns the automorphism group of the given linear code. The time needed by the implementation of the algorithm is comparable to Leon's program [10] for the calculation of the linear automorphism group of a linear code, but it additionally provides a unique representative and the automorphism group with respect to the more general notion of semilinear equivalence. The program can be used online under http://www.algorithm.uni-bayreuth.de/en/research/Coding_Theory/CanonicalForm/index.html.

Toric Legendrian Subvarieties

We give the full classification of smooth toric Legendrian subvarieties in P2n-1. We also prove that under some minor assumptions the group of linear automorphisms preserving a given Legendrian subvariety preserves the contact structure of the ambient projective space.

Laplacian Operators and Radon Transforms on Grassmann Graphs

Let Ω be a vector space over a finite field with q elements. Let G denote the general linear group of automorphisms of Ω and let us consider the left regular representation $\rho: {\rm G} \rightarrow {\cal B}(L_2({\rm X}))$ associated with the natural action of G on the set X of linear subspaces of Ω. In this paper we study a natural basis B of the algebra EndG(L 2(X)) of intertwining maps on L 2(X). By using a Laplacian operator on Grassmann graphs, we identify the kernels in B as solutions of a basic hypergeometric difference equation. This provides two expressions for these kernels. One in terms of the q-Hahn polynomials and the other by means of a Rodrigues type formula. Finally, we obtain a useful product formula for the mappings in B. We give two different proofs. One uses the theory of classical hypergeometric polynomials and the other is supported by a characterization of spherical functions in finite symmetric spaces. Both proofs require the use of certain associated Radon transforms.

Quartic K3 surfaces without nontrivial automorphisms

For any field k we fix an algebraic closure of k, denoted by k. For a variety X ⊂ P k we set X = X×k k, we say that X is smooth if X is regular, and we let AutX denote the group of k-automorphisms of X, while LinX denotes the group of linear automorphisms of X, i.e., automorphisms induced by a linear transformation of the coordinates of P. The following theorem was proved by Poonen, see [Po], Thm. 1.6.

Automorphisms of Tensor Completions of Algebras

In the classical representation of different groups, frequent use is made of a linear automorphism group of various algebras. Since the linear automorphism group is only part of a full automorphism group, such an approach might seem to be too restrictive. In this connection, we point out a natural, wide class of algebras whose automorphisms are standard and are reducible to linear. Thus, for algebras in this class, studying the full automorphism group reduces to treating the linear, a traditional approach in the class of such algebras being quite general.

Algebra Structure on the Hochschild Cohomology of the Ring of Invariants of a Weyl Algebra under a Finite Group

Let An be the nth Weyl algebra, and let G⊂Sp2n(C)⊂Aut(An) be a finite group of linear automorphisms of An. In this paper, we compute the multiplicative structure on the Hochschild cohomology HH•(AGn) of the algebra of invariants of G. We prove that, as a graded algebra, HH•(AGn) is isomorphic to the graded algebra associated to the center of the group algebra CG with respect to a filtration defined in terms of the defining representation of G.

Vector valued Riesz distributions on Euclidian Jordan algebras

Let V be a Euclidean Jordan algebra, Гthe associated symmetric cone and G be the identity component of the linear automorphism group of Г.In this paper we associate to a certain class of spherical representations (ρ, ɛ) of G certain ɛ-valued Riesz distributions generalizing the classical scalar valued Riesz distributions on V. Our construction is motivated by the analytic theory of unitary highest weight representations where it permits to study certain holomorphic families of operator valued Riesz distributions whose positive definiteness corresponds to the unitarity of a representation of the automorphism group of the associated tube domain Г +iV.

Double-(2n + 1) Configurations in PG(2n + 1, 2)

In this paper we establish the existence of a configuration in PG(2n+1,2),n≥2, a particular case of which is described in detail in [3]. The general configuration consists of two sets of 2n+1 spaces of dimensionnrelated by a bijection so that two relatedn-spaces meet in an (n−1)-space, two unrelatedn-spaces from different sets meet in a point and distinctn-spaces from the same set are disjoint. We determine the full group of linear automorphisms of these configurations and show that, for eachn≥3, those in PG(2n+1,2) are not all projectively equivalent to one another.

On fuzzy linear representations of finite groups

A linear representation ϱ of a finite group G in a finite-dimensional vector space V induces, through Zadeh's extension principle, a function, 9≈, from I G to into I GL( V) where GL( V) is the group of all linear automorphisms of V. If W is a fuzzy subspace of V, the group of all fuzzy linear automorphisms of W GL( W ), is a subgroup of GL( V). W is said to be stable under the action of a fuzzy subgroup A of G if 9≈( A ) is a subset of GL( W ), i.e. 9≈( A ) is zero at every f inside GL( V) and outside GL( q ). If W is stable under the action of A then its support subspace is stable under the action of the support subgroup of A in the crisp sense. Finally, we show that; if there are two stable fuzzy subspaces one of them is contained in the other, then the smaller one has a fuzzy direct summand in the bigger which is also a stable fuzzy subspace.