scholarly journals On orbits of the automorphism group on an affine toric variety

2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Ivan Arzhantsev ◽  
Ivan Bazhov
Keyword(s):  
Automorphism Group ◽  
Vector Space ◽  
Toric Variety ◽  
Connected Component ◽  
Linear Action

AbstractLet X be an affine toric variety. The total coordinates on X provide a canonical presentation $$\bar X \to X$$ of X as a quotient of a vector space $$\bar X$$ by a linear action of a quasitorus. We prove that the orbits of the connected component of the automorphism group Aut(X) on X coincide with the Luna strata defined by the canonical quotient presentation.

scholarly journals Automorphism groups of some variants of lattices

2021 ◽  
Vol 13 (1) ◽  
pp. 142-148
Author(s):  
O.G. Ganyushkin ◽  
O.O. Desiateryk
Keyword(s):  
Automorphism Group ◽  
Vector Space ◽  
Wreath Product ◽  
Automorphism Groups ◽  
Symmetric Groups ◽  
Natural Generalization ◽  
Space Lattice ◽  
Finite Vector ◽  
Finite Vector Space ◽  
Finite Set

In this paper we consider variants of the power set and the lattice of subspaces and study automorphism groups of these variants. We obtain irreducible generating sets for variants of subsets of a finite set lattice and subspaces of a finite vector space lattice. We prove that automorphism group of the variant of subsets of a finite set lattice is a wreath product of two symmetric permutation groups such as first of this groups acts on subsets. The automorphism group of the variant of the subspace of a finite vector space lattice is a natural generalization of the wreath product. The first multiplier of this generalized wreath product is the automorphism group of subspaces lattice and the second is defined by the certain set of symmetric groups.

Finite Groups the Centralizers of whose involutions have Normal 2-Complements

1969 ◽  
Vol 21 ◽  
pp. 335-357 ◽  
Author(s):  
Daniel Gorenstein
Keyword(s):  
Automorphism Group ◽  
Vector Space ◽  
Finite Groups ◽  
Classification Theorem ◽  
Natural Vector ◽  
Index 2

In this paper we shall classify all finite groups in which the centralizer of every involution has a normal 2-complement. For brevity, we call such a group an I-group. To state our classification theorem precisely, we need a preliminary definition.As is well-known, the automorphism group G = PΓL(2, q) of H= PSL(2, q), q= pn, is of the form G = LF, where L = PGL(2, q), L ⨞ G, F is cyclic of order n, L ∩ F = 1, and the elements of F are induced from semilinear transformations of the natural vector space on which GL(2, q) acts; cf. (3, Lemma 2.1) or (7, Lemma 3.3). It follows at once (4, Lemma 2.1; 8, Lemma 3.1) that the groups H and L are each I-groups. Moreover, when q is an odd square, there is another subgroup of G in addition to L that contains H as a subgroup of index 2 and which is an I-group.

scholarly journals Automorphisms of Drinfeld half-spaces over a finite field

2013 ◽  
Vol 149 (7) ◽  
pp. 1211-1224 ◽  
Author(s):  
Bertrand Rémy ◽  
Amaury Thuillier ◽  
Annette Werner
Keyword(s):  
Automorphism Group ◽  
Finite Field ◽  
Vector Space ◽  
Half Space ◽  
Linear Group ◽  
Analytic Geometry ◽  
Base Field ◽  
Projective Linear Group

AbstractWe show that the automorphism group of Drinfeld’s half-space over a finite field is the projective linear group of the underlying vector space. The proof of this result uses analytic geometry in the sense of Berkovich over the finite field equipped with the trivial valuation. We also take into account extensions of the base field.

scholarly journals On the component group of the automorphism group of a Lie group

1994 ◽  
Vol 57 (3) ◽  
pp. 365-385 ◽  
Author(s):  
P. B. Chen ◽  
T. S. Wu
Keyword(s):  
Automorphism Group ◽  
Lie Group ◽  
Faithful Representation ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Connected Component ◽  
Component Group ◽  
Necessary And Sufficient

AbstractLet G be a Lie group, Go the connected component of G that contains the identity, and Aut G the group of all topological automorphisms of G. In the case when G/Go is finite and G has a faithful representation, we obtain a necessary and sufficient condition for G so that Aut G has finitely many components in terms of the maximal central torus in Go.

scholarly journals Box splines and the equivariant index theorem

2012 ◽  
Vol 12 (3) ◽  
pp. 503-544 ◽  
Author(s):  
C. De Concini ◽  
C. Procesi ◽  
M. Vergne
Keyword(s):  
Vector Space ◽  
Elliptic Operator ◽  
Inversion Formula ◽  
Equivariant Cohomology ◽  
Index Theorem ◽  
Vector Spaces ◽  
Box Splines ◽  
Linear Action ◽  
Compact Torus ◽  
Box Spline

AbstractIn this article, we begin by recalling the inversion formula for the convolution with the box spline. The equivariant cohomology and the equivariant $K$-theory with respect to a compact torus $G$ of various spaces associated to a linear action of $G$ in a vector space $M$ can both be described using some vector spaces of distributions, on the dual of the group $G$ or on the dual of its Lie algebra $\mathfrak{g}$. The morphism from $K$-theory to cohomology is analyzed, and multiplication by the Todd class is shown to correspond to the operator (deconvolution) inverting the semi-discrete convolution with a box spline. Finally, the multiplicities of the index of a $G$-transversally elliptic operator on $M$ are determined using the infinitesimal index of the symbol.

Automorphism Group of a Class of Gradation Shifting Toroidal Lie Algebras

2010 ◽  
Vol 17 (04) ◽  
pp. 705-720 ◽  
Author(s):  
Zhangsheng Xia ◽  
Shaobin Tan ◽  
Haifeng Lian
Keyword(s):  
Automorphism Group ◽  
Vector Space ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Linear Groups ◽  
Type B ◽  
Laurent Polynomials ◽  
Lie Bracket ◽  
Toroidal Algebra ◽  
Toroidal Lie Algebra

Let [Formula: see text] be the ring of Laurent polynomials in commuting variables. As a generalization of the toroidal Lie algebra, the gradation shifting toroidal Lie algebra [Formula: see text] is isomorphic to the corresponding (centerless) toroidal Lie algebra so(n, ℂ) ⨂ A of type B or D as a vector space, with the Lie bracket twisted by n fixed elements E1,…,En from A. In this paper, we study the automorphisms of the gradation shifting toroidal algebra [Formula: see text], which is proved to be closely related to a class of subgroups of GL(n,ℤ), called the linear groups over semilattices. We use the linear group over a special semilattice to determine the automorphism group of the gradation shifting toroidal algebra [Formula: see text], which extends our earlier work.

scholarly journals Automorphisms and isomorphisms of enhanced hypercubes

Filomat ◽  
2020 ◽  
Vol 34 (8) ◽  
pp. 2805-2812
Author(s):  
Lu Lu ◽  
Qiongxiang Huang
Keyword(s):  
Automorphism Group ◽  
Vector Space ◽  
Cayley Graph ◽  
Cayley Graphs ◽  
Standard Basis ◽  
Generating Set ◽  
Folded Hypercube

Let Zn2 be the elementary abelian 2-group, which can be viewed as the vector space of dimension n over F2. Let {e1,..., en} be the standard basis of Zn2 and ?k = ek +...+ en for some 1 ? k ? n-1. Denote by ?n,k the Cayley graph over Zn2 with generating set Sk = {e1,..., en,?k}, that is, ?n,k = Cay(Zn2,Sk). In this paper, we characterize the automorphism group of ?n,k for 1 ? k ? n-1 and determine all Cayley graphs over Zn2 isomorphic to ?n,k. Furthermore, we prove that for any Cayley graph ? = Cay(Zn2,T), if ? and ?n,k share the same spectrum, then ? ? ?n,k. Note that ?n,1 is known as the so called n-dimensional folded hypercube FQn, and ?n,k is known as the n-dimensional enhanced hypercube Qn,k.

scholarly journals The automorphism group and fixing number of orthogonality graph over a vector space

2021 ◽  
pp. 2350013
Author(s):  
Shikun Ou ◽  
Yanfei Tan
Keyword(s):  
Automorphism Group ◽  
Vector Space ◽  
Simple Graph ◽  
Metric Dimension ◽  
Vertex Set ◽  
Field Automorphism

Let [Formula: see text] be a field, and [Formula: see text] the [Formula: see text]-dimensional row vector space over [Formula: see text]. The orthogonality graph [Formula: see text] of [Formula: see text] is an undirected simple graph which has [Formula: see text] as its vertex set, and for distinct [Formula: see text], [Formula: see text] if and only if [Formula: see text], where [Formula: see text] is the transpose of [Formula: see text]. When [Formula: see text] is finite, it is shown that any automorphism of [Formula: see text] can be decomposed into the product of a row-orthogonal automorphism and either a permutation automorphism or a field automorphism; moreover, the fixing number and metric dimension of [Formula: see text] are considered.

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