Orthogonal Group Research Articles

Representation of coxeter group and orthogonal group

The paper is primarily divided into two parts. The main focus of the first part is the construction of a representation of Coxeter groups. This begins with the definition of the Coxeter system and connected components, followed by the introduction of the length function and subsequent theorems. The faithfulness of this representation is then proven, allowing for the identification of isomorphisms that enable the final classification of finite Coxeter groups. This classification is achieved by leveraging the established relationship between irreducible representations of Coxeter groups and positive definite quadratic forms. Given the strong connection between Coxeter groups and orthogonal groups, the primary objective of the second part is to create a specific representation of orthogonal groups. This is accomplished through an examination of the decomposition of harmonic polynomials into subspaces of homogeneous harmonic polynomials, using the action of O(2) on these subspaces. The paper concludes by drawing connections to results in Invariant Theory, demonstrating the applicability of the presented concepts in a more general duality context.

On Oliver's p-group conjecture for Sylow subgroups of symplectic groups and orthogonal groups

In this paper, we focus on Oliver's p-group conjecture. We prove that Oliver's p-group conjecture holds for Sylow p-subgroups of symplectic groups and orthogonal groups.

Orbits in lattices

Let L be a lattice. We exhibit algorithms for calculating Tits buildings and orbits of vectors in L for certain subgroups of the orthogonal group O(L). We discuss how these algorithms can be applied to determine the configuration of boundary components in the Baily-Borel compactification of orthogonal modular varieties and to improve the performance of computer arithmetic of orthogonal modular forms.

Characterising the Haar measure on the [formula omitted]-adic rotation groups via inverse limits of measure spaces

We determine the Haar measure on the compact p-adic special orthogonal groups of rotations SO(d)p in dimension d=2,3, by exploiting the machinery of inverse limits of measure spaces, for every prime p>2. We characterise the groups SO(d)p as inverse limits of finite groups, of which we provide parametrisations and orders, together with an equivalent description through a multivariable Hensel lifting. Supplying these finite groups with their normalised counting measures, we get an inverse family of Haar measure spaces for each SO(d)p. Finally, we constructively prove the existence of the so-called inverse limit measure of these inverse families, which is explicitly computable, and prove that it gives the Haar measure on SO(d)p. Our results pave the way towards the study of the irreducible projective unitary representations of the p-adic rotation groups, with potential applications to the recently proposed p-adic quantum information theory.

On normalizers of maximal tori in classical Lie groups

In general, the normalizer N G ( H G ) N_G(H_G) of a maximal torus H G H_G in a semisimple complex Lie group G G does not admit a presentation as a semidirect product of H G H_G and the corresponding Weyl group W G W_G . Meanwhile, such splitting exists for classical groups corresponding to the root systems A ℓ A_\ell , B ℓ B_\ell , D ℓ D_\ell . For the remaining classical groups corresponding to the root systems C ℓ C_\ell , there still exists an embedding of the Tits extension W G T W^T_G of W G W_G into the normalizer N G ( H G ) N_G(H_G) . Here an explicit unified construction is proposed for the lifts of the Weyl groups into normalizers of maximal tori for general linear and orthogonal Lie groups via embedding into the general linear groups. For the symplectic group S p 2 ℓ \mathrm {Sp}_{2\ell } , an explanation of the impossibility of embedding W S p 2 ℓ W_{\mathrm {Sp}_{2\ell }} into the normalizer N S p 2 ℓ ( H S p 2 ℓ ) N_{\mathrm {Sp}_{2\ell }}(H_{\mathrm {Sp}_{2\ell }}) is suggested. Also, explicit formulas are computed for the adjoint action of the lifts of the Weyl groups on g = L i e ( G ) \mathfrak {g}=\mathrm {Lie}(G) . Finally, examples of Weyl groups lifts are given for the groups closely associated with classical Lie groups.

Products of unipotent elements of index 2 in orthogonal and symplectic groups

An automorphism u of a vector space is called unipotent of index 2 whenever (u−id)2=0. Let b be a non-degenerate symmetric or skewsymmetric bilinear form on a vector space V over a field F of characteristic different from 2.Here, we characterize the elements of the isometry group of b that are the product of two unipotent isometries of index 2. In particular, if b is skewsymmetric and nondegenerate we prove that an element of the symplectic group of b is the product of two unipotent isometries of index 2 if and only if it has no Jordan cell of odd size for the eigenvalue −1. As an application, we prove that every element of a symplectic group is the product of three unipotent elements of index 2 (and no fewer in general).For orthogonal groups, the classification closely matches the classification of sums of two square-zero skewselfadjoint operators that was obtained in a recent article [7].

Invariant measures on p-adic Lie groups: the p-adic quaternion algebra and the Haar integral on the p-adic rotation groups

We provide a general expression of the Haar measure—that is, the essentially unique translation-invariant measure—on a p-adic Lie group. We then argue that this measure can be regarded as the measure naturally induced by the invariant volume form on the group, as it happens for a standard Lie group over the reals. As an important application, we next consider the problem of determining the Haar measure on the p-adic special orthogonal groups in dimension two, three and four (for every prime number p). In particular, the Haar measure on SO(2,Qp)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {SO}(2,\\mathbb {Q}_p)$$\\end{document} is obtained by a direct application of our general formula. As for SO(3,Qp)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {SO}(3,\\mathbb {Q}_p)$$\\end{document} and SO(4,Qp)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {SO}(4,\\mathbb {Q}_p)$$\\end{document}, instead, we show that Haar integrals on these two groups can conveniently be lifted to Haar integrals on certain p-adic Lie groups from which the special orthogonal groups are obtained as quotients. This construction involves a suitable quaternion algebra over the field Qp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Q}_p$$\\end{document} and is reminiscent of the quaternionic realization of the real rotation groups. Our results should pave the way to the development of harmonic analysis on the p-adic special orthogonal groups, with potential applications in p-adic quantum mechanics and in the recently proposed p-adic quantum information theory.

Howe correspondence of unipotent characters for a finite symplectic/even-orthogonal dual pair

abstract: In this paper we give a complete and explicit description of the Howe correspondence of unipotent characters for a finite reductive dual pair of a symplectic group and an even orthogonal group in terms of the Lusztig parametrization. That is, the conjecture by Aubert-Michel-Rouquier is confirmed.

K3 surfaces, cyclotomic polynomials and orthogonal groups

Let X be a complex projective K3 surface and let TX\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T_X$$\\end{document} be its transcendental lattice; the characteristic polynomials of isometries of TX\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T_X$$\\end{document} induced by automorphisms of X are powers of cyclotomic polynomials. Which powers of cyclotomic polynomials occur? The aim of this note is to answer this question, as well as related ones, and give an alternative approach to some results of Kondō, Machida, Oguiso, Vorontsov, Xiao and Zhang; this leads to questions and results concerning orthogonal groups of lattices.

Rotationally invariant translators of the mean curvature flow in Einstein's static universe

In this paper, we deal with non-degenerate translators of the mean curvature flow in the well-known Einstein's static universe. We focus on the rotationally invariant translators, that is, those invariant by a natural isometric action of the special orthogonal group on the ambient space. In the classification list, there are three space-like cases and five time-like cases. All of them, except a totally geodesic example, have one or two conic singularities. Also, we show a uniqueness result based on the behaviour of the translator on its boundary. As an application, we extend an isometry of the sphere to the whole translator under simple conditions. This leads to a characterization of a bowl-like example.

The Endoscopic Classification of Representations of Non-Quasi-Split Odd Special Orthogonal Groups

Abstract In an earlier book of Arthur, the endoscopic classification of representations of quasi-split orthogonal and symplectic groups was established. Later Mok gave that of quasi-split unitary groups. After that, Kaletha, Minguez, Shin, and White gave that of non-quasi-split unitary groups for generic parameters. In this paper we prove the endoscopic classification of representations of non-quasi-split odd special orthogonal groups for generic parameters, following Kaletha, Minguez, Shin, and White.

Factoring determinants and applications to number theory

Products of shifted characteristic polynomials, and ratios of such products, averaged over the classical compact groups are of great interest to number theorists as they model similar averages of [Formula: see text]-functions in families with the same symmetry type as the compact group. We use Toeplitz and Toeplitz plus Hankel operators and the identities of Borodin–Okounkov–Case–Geronimo, and Basor–Ehrhardt to prove that, in certain cases, these unitary averages factor as polynomials into averages over the symplectic group and the orthogonal group. Building on these identities we present new proofs of the exact formulas for these averages where the “swap” terms that are characteristic of the number theoretic averages occur from the Fredholm expansions of the determinants of the appropriate Hankel operator. This is the fourth different proof of the formula for the averages of ratios of products of shifted characteristic polynomials; the other proofs are based on supersymmetry; symmetric function theory, and orthogonal polynomial methods from random matrix theory.

Integrated Models of Cleaner Production Technologies for Maize Cultivation in China’s Black Soil Regions

Incorporating carbon footprints into the production efficiency framework to construct a standardized technology model for cleaner production in black soil regions is of great significance for improving the soil environment and the sustainable development of agriculture. This study used an orthogonal experimental design and the DEA–Malmquist method to calculate the carbon footprint and total factor productivity of orthogonal experimental groups of cleaner production technologies for maize cultivation in China’s black soil region and then identified integrated models of cleaner production technology in the black soil region. The results showed that the carbon footprint of maize cultivation and total factor productivity were generally higher in the experimental group using cleaner production techniques than in the control group. Still, none of them reached the optimum. In the future, the synergistic effect of technological progress and technological efficiency enhancement should be brought into play, and the integrated models of “Soil testing and formulation + Full mobile sprinkler irrigation + Straw tilling and field return” and “No tillage in spring + Soil testing and formulation + Straw tilling and field return” should be promoted in semi-arid and semi-humid black soil regions, which can improve the low carbon productivity of maize by 20.3% and 15.4%, respectively.

Tunable magnetism in titanium-based kagome metals by rare-earth engineering and high pressure

Rare-earth engineering is an effective way to introduce and tune magnetism in topological kagome materials, which have been acting as a fertile platform to investigate the quantum interactions between geometry, topology, spin, and correlation. Here, we report the synthesis, structure, and physical properties of titanium-based kagome metals RETi3Bi4 (RE = Yb, Pr, and Nd) with various magnetic states. They all crystallize in the orthogonal space group Fmmm (No. 69), featuring distorted titanium kagome lattices and rare-earth zig-zag chains. By changing the rare earth atoms in the zig-zag chains, the magnetism can be tuned from nonmagnetic YbTi3Bi4 to short-range ordered PrTi3Bi4 (Tanomaly ~ 8.2 K), and finally to ferromagnetic NdTi3Bi4 (Tc ~ 8.5 K). In-situ resistance measurements of NdTi3Bi4 under high pressure further reveal a tunable ferromagnetic ordering temperature. These results highlight RETi3Bi4 as a promising family of kagome metals to explore nontrivial band topology and exotic phases.

On the Structure of SO(3): Trace and Canonical Decompositions

This paper is devoted to some selected topics of the theory of special orthogonal group SO(3). First, we discuss the trace of orthogonal matrices and its relation to the characteristic polynomial; on this basis, the partition of SO(3) formed by conjugation classes is described by trace mapping. Second, we show that every special orthogonal matrix can be expressed as the product of three elementary special orthogonal matrices. Explicit formulas for the decomposition as needed for applications in differential geometry and physics as symmetry transformations are given.

Arthur Packets for Quasisplit GSp(2n) and GO(2n) Over a p-Adic Field

Abstract We construct the Arthur packets for symplectic and even orthogonal similitude groups over a $p$-adic field and show that they are stable and satisfy the twisted endoscopic character relations.

Large deviations for random matrices in the orthogonal group and Stiefel manifold with applications to random projections of product distributions

Large deviations for random matrices in the orthogonal group and Stiefel manifold with applications to random projections of product distributions

Hecke theory for SO+(2,n + 2)

We describe the foundations of a Hecke theory for the orthogonal group SO+(2,n+2). In particular we consider the Hermitian modular group of degree 2 as a special example of SO+(2,4). As an application we show that the attached Maaß space is invariant under Hecke operators. This implies that the Eisenstein series belongs to the Maaß space. If the underlying lattice is even and unimodular, our approach allows us to reprove the explicit formula of its Fourier coefficients.

$SO(3)$ квазі-мономіальні сім'ї многочленів

Let $H$ be a subgroup of the affine space group ${\rm Aff} 3)$, considered with its natural action on the vector space of three-variable polynomials. The polynomial family $\{ B_{m,n,k}(x,y,z) \}$ is called quasi-monomial with respect to $H$ if the group operators in two different bases $\{ x^m y^n z^k \}$ and $\{ B_{m,n,k}(x,y,z) \}$ have identical atrices. We derive a criterion for quasi-monomiality when the group $H$ is the special orthogonal group $SO(3)$. This criterion is expressed through the exponential generating function of the polynomial family $\{B_{m,n,k}(x,y,z)\}$. It has been proven that Appel's biorthogonal polynomials are quasi-monomials with respect to $SO(3)$ and recurrence relations have been found for them.

Leibniz cohomology with adjoint coefficients

With the Poincaré group R 3 , 1 ⋊ O ( 3 , 1 ) as the model of departure, we focus, for n = p + q , p + q ≥ 4 , on the affine indefinite orthogonal group R p , q ⋊ SO ( p , q ) . Denote by h p , q the Lie algebra of the affine indefinite orthogonal group. We compute the Leibniz cohomology of h p , q with adjoint coefficients, written H L * ( h p , q ; h p , q ) . We calculate several indefinite orthogonal invariants, and h p , q -invariants.