Fixed Point Group Research Articles

Fixed point groups of involutions of type O(q,k) over a field of characteristic two

ABSTRACT For G = O ⁡ ( q , k ) , the orthogonal group over a field k of characteristic 2 with respect to a quadratic form q , we discuss the G-conjugacy classes of fixed points of involutions. When the quadratic space is either totally singular or nonsingular, a full classification of the isomorphism classes is given. We also give some implications of these results for a general quadratic space over a field of characteristic 2.

Chabauty limits of groups of involutions In SL(2,F) for local fields

We classify Chabauty limits of groups fixed by various (abstract) involutions over SL ( 2 , F ) , where F is a finite field-extension of Q p , with p ≠ 2 . To do so, we first classify abstract involutions over SL ( 2 , F ) with F a quadratic extension of Q p , and prove p-adic polar decompositions with respect to various subgroups of p-adic SL 2 . Then we classify Chabauty limits of: SL ( 2 , F ) ⊂ SL ( 2 , E ) where E is a quadratic extension of F, of SL ( 2 , R ) ⊂ SL ( 2 , C ) , and of H θ ⊂ SL ( 2 , F ) , where H θ is the fixed point group of an F-involution θ over SL ( 2 , F ) .

Fixed-point group conjugacy classes of unipotent elements in low-dimensional symmetric spaces of special linear groups over a finite field

In this paper, we characterize and classify the orbits of the fixed-point group on the unipotent elements of the generalized symmetric spaces for inner involutions of [Formula: see text] and [Formula: see text] where [Formula: see text] is a finite field of odd characteristic. We provide some generalized results for [Formula: see text].

Extended Bound States in the Continuum with Symmetry‐Broken Terahertz Dielectric Metasurfaces

AbstractExcitation of the non‐radiative eigenmodes in photonics and metamaterials above their light cone, that is, the bound states in the continuum (BICs), has drawn fundamental and technological interest. However, the current studies on photonic BICs are mainly restricted to the characterization of the eigenmodes with fixed point group. Here, a fourfold rotationally symmetric (C4v) array of silicon resonators in a square lattice is investigated that reveals the traditional symmetry‐protected (Γ‐state) and accidental (off‐Γ) BICs in the momentum space. Upon symmetry‐breaking, BICs at Γ‐state transform into quasi‐states with decaying quality factors that obey the inverse‐square law α−2 as a function of the structural asymmetry parameter α. Surprisingly, at larger asymmetries, multiple new states of extended BICs are observed beyond typical inverse‐square law behavior. The structural asymmetry of a resonator provides an important degree of freedom for exploring the Friedrich–Wintgen and new accidental BICs that are topologically protected by robust polarization vortices. The extended family of sharp BIC resonances in all‐dielectric silicon metasurfaces is extremely important for scalable applications in terahertz sensing, higher harmonic generation, and ultralow‐power switching.

Minimal parabolic k-subgroups acting on symmetric k-varieties corresponding to k-split groups

Symmetric [Formula: see text]-varieties are a natural generalization of symmetric spaces to general fields [Formula: see text]. We study the action of minimal parabolic [Formula: see text]-subgroups on symmetric [Formula: see text]-varieties and define a map that embeds these orbits within the orbits corresponding to algebraically closed fields. We develop a condition for the surjectivity of this map in the case of [Formula: see text]-split groups that depends only on the dimension of a maximal [Formula: see text]-split torus contained within the fixed point group of the involution defining the symmetric [Formula: see text]-variety.

Classifying the orbits of the generalized symmetric spaces for

In this paper we will discuss the orbits of the fixed-point group on the tori of the generalized symmetric spaces of where k is a finite field. Specifically, we will provide a characterization and classification of the maximal k-split and k-anisotropic tori.

PARABOLIC COMPACTIFICATION OF HOMOGENEOUS SPACES

AbstractIn this article, we study compactifications of homogeneous spaces coming from equivariant, open embeddings into a generalized flag manifold $G/P$. The key to this approach is that in each case $G/P$ is the homogeneous model for a parabolic geometry; the theory of such geometries provides a large supply of geometric tools and invariant differential operators that can be used for this study. A classical theorem of Wolf shows that any involutive automorphism of a semisimple Lie group $G$ with fixed point group $H$ gives rise to a large family of such compactifications of homogeneous spaces of $H$. Most examples of (classical) Riemannian symmetric spaces as well as many non-symmetric examples arise in this way. A specific feature of the approach is that any compactification of that type comes with the notion of ‘curved analog’ to which the tools we develop also apply. The model example of this is a general Poincaré–Einstein manifold forming the curved analog of the conformal compactification of hyperbolic space. In the first part of the article, we derive general tools for the analysis of such compactifications. In the second part, we analyze two families of examples in detail, which in particular contain compactifications of the symmetric spaces $\mathit{SL}(n,\mathbb{R})/\mathit{SO}(p,n-p)$ and $\mathit{SO}(n,\mathbb{C})/\mathit{SO}(n)$. We describe the decomposition of the compactification into orbits, show how orbit closures can be described as the zero sets of smooth solutions to certain invariant differential operators and prove a local slice theorem around each orbit in these examples.

The classification of involutions and symmetric spaces of modular groups

The involutions and the symmetric spaces associated to the family of modular groups of order 2m are explored. We begin by analyzing the structure of the automorphism group and by establishing which automorphisms are involutions. We conclude by calculating the fixed-point group and symmetric spaces determined by each involution.

Coprime actions with supersolvable fixed-point groups

Abstract Let A be an elementary abelian r-group acting on a finite r ′ {r^{\prime}} -group G. Suppose that the fixed-point group C G ⁡ ( a ) {\operatorname{C}_{G}(a)} is supersolvable for each a ∈ A # {a\in A^{\#}} . We show that G is supersolvable if | A | ⩾ r 4 {|A|\geqslant r^{4}} and that G ′ ⩽ 𝐅 3 ⁢ ( G ) {G^{\prime}\leqslant\mathbf{F}_{3}(G)} if | A | ⩾ r 3 {|A|\geqslant r^{3}} . Moreover, we prove some other results for cases when the fixed-point group C G ⁡ ( a ) {\operatorname{C}_{G}(a)} is abelian, p-nilpotent or satisfies the Sylow tower property.

K-Involutions of SL(n,k) over fields of characteristic 2

ABSTRACTSymmetric k-varieties generalize Riemannian symmetric spaces to reductive groups defined over arbitrary fields. For most perfect fields, it is known that symmetric k-varieties are in one-to-one correspondence with isomorphy classes of k-involutions. Therefore, it is useful to have representatives of each isomorphy class in order to describe the k-varieties. Here we give matrix representatives for each isomorphy class of k-involutions of SL(n,k) in the case that k is any field of characteristic 2; we also describe fixed-point groups of each type of involution.

Involutions of Type G2 Over Fields of Characteristic Two

We continue a study of automorphisms of order 2 of algebraic groups. In particular we look at groups of type G2 over fields k of characteristic two. Let C be an octonion algebra over k; then Aut(C) is a group of type G2 over k. We characterize automorphisms of order 2 and their corresponding fixed point groups for Aut(C) by establishing a connection between the structure of certain four dimensional subalgebras of C and the elements in Aut(C) that induce inner automorphisms of order 2. These automorphisms relate to certain quadratic forms which, in turn, determine the Galois cohomology of the fixed point groups of the involutions. The characteristic two case is unique because of the existence of four dimensional totally singular subalgebras. Over finite fields we show how our results coincide with known results, and we establish a classification of automorphisms of order 2 over infinite fields of characteristic two.

Lifting Representations of Finite Reductive Groups I: Semisimple Conjugacy Classes

AbstractSuppose that is a connected reductive group defined over a field k, and ┌ is a finite group acting via k-automorphisms of satisfying a certain quasi-semisimplicity condition. Then the identity component of the group of -fixed points in is reductive. We axiomatize the main features of the relationship between this fixed-point group and the pair (,┌) and consider any group G satisfying the axioms. If both and G are k-quasisplit, then we can consider their duals *and G*. We show the existence of and give an explicit formula for a natural map from the set of semisimple stable conjugacy classes in G*(k) to the analogous set for *(k). If k is finite, then our groups are automatically quasisplit, and our result specializes to give a map of semisimple conjugacy classes. Since such classes parametrize packets of irreducible representations of G(k) and (k), one obtains a mapping of such packets.

Isomorphy classes of k-involutions of G2

Isomorphy classes of k-involutions have been studied for their correspondence with symmetric k-varieties, also called generalized symmetric spaces. A symmetric k-variety of a k-group G is defined as Gk/Hk where θ : G → G is an automorphism of order 2 that is defined over k and Gk and Hk are the k-rational points of G and H = Gθ, the fixed point group of θ, respectively. This is a continuation of papers written by A. G. Helminck and collaborators [Involutions of SL (2, k), (n > 2), Acta Appl. Math.90(1–2) (2006) 91–119, Classification of involutions of SO (n; k; b), to appear, On the classification of k-involutions, Adv. Math.153(1) (1988) 1–117, Classification of involutions of SL (2, k), Comm. Algebra30(1) (2002) 193–203] expanding on his combinatorial classification over certain fields. Results have been achieved for groups of type A, B and D. Here we begin a series of papers doing the same for algebraic groups of exceptional type.

Constant term of Eisenstein integrals on a reductive 𝑝-adic symmetric space

Let H H be the fixed point group of a rational involution σ \sigma of a reductive p p -adic group on a field of characteristic different from 2. Let P P be a σ \sigma -parabolic subgroup of G G , i.e. such that σ ( P ) \sigma (P) is opposite P P . We denote the intersection P ∩ σ ( P ) P\cap \sigma (P) by M M . Kato and Takano on one hand and Lagier on the other associated canonically to an H H -form, i.e. an H H -fixed linear form, ξ \xi , on a smooth admissible G G -module, V V , a linear form on the Jacquet module j P ( V ) j_P(V) of V V along P P which is fixed by M ∩ H M\cap H . We call this operation the constant term of H H -forms. This constant term is linked to the asymptotic behaviour of the generalized coefficients with respect to ξ \xi . P. Blanc and the second author defined a family of H H -forms on certain parabolically induced representations, associated to an M ∩ H M\cap H -form, η \eta , on the space of the inducing representation. The purpose of this article is to describe the constant term of these H H -forms. Also it is shown that when η \eta is discrete, i.e. when the generalized coefficients of η \eta are square integrable modulo the center, the corresponding family of H H -forms on the induced representations is a family of tempered, in a suitable sense, H H -forms. A formula for the constant term of Eisenstein integrals is given.

On lower bounds for cohomology growth in $$p$$ p -adic analytic towers

Let $$p$$ and $$\ell $$ be two distinct prime numbers and let $$\Gamma $$ be a group. We study the asymptotic behaviour of the mod- $$\ell $$ Betti numbers in $$p$$ -adic analytic towers of finite index subgroups. If $$\Theta $$ is a finite $$\ell $$ -group of automorphisms of $$\Gamma $$ , our main theorem allows to lift lower bounds for the mod- $$\ell $$ cohomology growth in the fixed point group $$\Gamma ^\Theta $$ to lower bounds for the growth in $$\Gamma $$ . We give applications to $$S$$ -arithmetic groups and we also obtain a similar result for cohomology with rational coefficients.

On generalized Cartan subspaces

Let G be a connected reductive algebraic group defined over a field k of characteristic not 2, θ an involution of G defined over k, H a k-open subgroup of the fixed point group of θ and Gk (resp. Hk) the set of k-rational points of G (resp. H). The variety Gk/Hk is called a symmetric k-variety. For real and p-adic symmetric k-varieties the space L2(Gk/Hk) of square integrable functions decomposes into several series, one for each Hk-conjugacy class of Cartan subspaces of Gk/Hk.

Fixed points of complex-valued bidirectional associative memory

A discrete complex-valued bidirectional associative memory (CVBAM) is considered. The input and output of CVBAM are bipolar complex numbers, of which the two components take the value of either −1 or 1. We prove that the stored patterns are all fixed points of CVBAM provided certain sufficient conditions are satisfied. It is shown that each fixed point belongs to a fixed point group which consists of four fixed points. Supporting numerical results are provided.

Decompositions related to symmetric varieties

Let G be a connected reductive group over an algebraically closed field of characteristic ≠2 and let σ be an automorphism of G of order 2, whose fixed point group is K. The paper establishes for certain G-varieties X a decomposition of X in finitely many K-stable locally closed pieces. Namely, (a) if X = G / P , where P is a parabolic subgroup of G and (b) ( G adjoint) if X is the wonderful compactification of the symmetric variety G / K .

On the Classification of Orbits of Symmetric Subgroups Acting on Flag Varieties of SL(2, k)

Symmetric k-varieties are a generalization of symmetric spaces to general fields. Orbits of a minimal parabolic k-subgroup acting on a symmetric k-variety are essential in the study of symmetric k-varieties and their representations. In this article, we present the classification of these orbits for the group SL(2,k) for a number of base fields k, including finite fields and the 𝔭-adic numbers. We use the characterization in Helminck and Wang (1993), which requires one to first classify the orbits of the θ-stable maximal k-split tori under the action of the k-points of the fixed point group.

On Galois theory from subfactors

We review some concepts of Galois theory for subfactors N ⊆ M , computing some Galois groups and correspondences in this framework. Given an outer action of a group G on a II 1 von Neumann factor N and a normal subgroup H of G, we prove that the fixed point group–subgroup subfactor N G ⊆ N H is conjugate to the group–subfactor N G / H ⊆ N .