Orthogonal Group Research Articles

Critical varieties and motivic equivalence for algebras with involution

Motivic equivalence for algebraic groups was recently introduced by De Clercq [Compos. Math. 153 (2017), pp. 2195–2213], where a characterization of motivic equivalent groups in terms of higher Tits indexes is given. As a consequence, if the quadrics associated to two quadratic forms have the same Chow motives with coefficients in F 2 \mathbb {F}_2 , this remains true for any two projective homogeneous varieties of the same type under the orthogonal groups of those two quadratic forms. Our main result extends this to all groups of classical type, and to some exceptional groups, introducing a notion of critical variety. On the way, we prove that motivic equivalence of the automorphism groups of two involutions can be checked after extending scalars to some index reduction field, which depends on the type of the involutions. In addition, we describe conditions on the base field which guarantee that motivic equivalent involutions actually are isomorphic, extending a result of Hoffmann on quadratic forms.

Divergent synthesis of biologically active L-threo-β-hydroxyaspartates from common trans-oxazolidine dicarboxylate.

A divergent synthetic strategy starting from a common trans-oxazolidine dicarboxylate intermediate has been successful to produce several non-proteinogenic L-threo-β-hydroxyaspartate derivatives efficiently with high stereoselectivity. Three bioactive α-amino-β-hydroxy acids, L-threo-β-hydroxyaspartic acid, L-threo-β-hydroxyasparagine, and L-threo-β-benzyloxyaspartic acid, were synthesized in good yields (58-83%) from the common chiral intermediate, and the chemoselective peptide bond formation at the α-amino group, β-hydroxy group, or α-carboxylic acid of the common intermediate was possible to afford the corresponding dipeptide, tripeptide, or didepsipeptide intermediate in 46~77% yields (in three-to-four steps) due to the orthogonal protective groups on the chiral intermediate.

Optimal orthogonal group synchronization and rotation group synchronization

Abstract We study the statistical estimation problem of orthogonal group synchronization and rotation group synchronization. The model is $Y_{ij} = Z_i^* Z_j^{*T} + \sigma W_{ij}\in{\mathbb{R}}^{d\times d}$ where $W_{ij}$ is a Gaussian random matrix and $Z_i^*$ is either an orthogonal matrix or a rotation matrix, and each $Y_{ij}$ is observed independently with probability $p$. We analyze an iterative polar decomposition algorithm for the estimation of $Z^*$ and show it has an error of $(1+o(1))\frac{\sigma ^2 d(d-1)}{2np}$ when initialized by spectral methods. A matching minimax lower bound is further established that leads to the optimality of the proposed algorithm as it achieves the exact minimax risk.

Geometry of the neutrino mixing space

We study the geometric structure of the physical region of neutrino mixing matrices as part of the unit ball of the spectral norm. Each matrix from the geometric region is a convex combination of unitary PMNS matrices. The disjoint subsets corresponding to a different minimal number of additional neutrinos are described as relative interiors of faces of the unit ball. We determined the Carath\'eodory's number showing that at most four unitary matrices of dimension three are necessary to represent any matrix from the neutrino geometric region. For matrices which correspond to scenarios with one and two additional neutrino states, the Carath\'eodory's number is two and three, respectively. Further, we discuss the volume associated with different mathematical structures, particularly with unitary and orthogonal groups, and the unit ball of the spectral norm. We compare the obtained volumes to the volume of the region of physically admissible mixing matrices for both the $CP$-conserving and $CP$-violating cases in the present scenario with three neutrino families and scenarios with the neutrino mixing matrix of dimension higher than three.

Distributed attitude tracking control of multiple rigid bodies with global exponential stability on SO(3)

AbstractIn this article, the leader‐following attitude tracking problem is considered for a group of rigid bodies, and a distributed control algorithm is proposed with global exponential stability on the rotational orthogonal group SO(3). In particular, we consider that only a subset of followers has access to the leader's trajectory, and a semi‐global exponential distributed observer is first proposed for each follower to estimate the leader's trajectory. Then, we introduce the shifted reference attitude of the leader to circumvent the topological obstruction on SO(3), and a combined distributed observer algorithm is developed by synthesizing different reference attitude trajectories. By constructing two Lyapunov functions, it is demonstrated that the leader's trajectory can be accurately estimated by the proposed distributed observer with exponential convergence. Next, an observer‐based attitude tracking control algorithm is synthesized, which is continuous in time but is discontinuous with respect to initial conditions. It is shown that the trajectory of the leader can be tracked by all followers with global exponential stability on SO(3). Finally, numerical simulation examples are provided to illustrate the effectiveness of the proposed control algorithms.

Field-Based Human-Centred Control on SO(3) for Assist-as-Needed Robotic Rehabilitation

Assist-as-needed (AAN) control can effectively improve the participation of the patient. However, the current research on attitude control in a manifold and also a special orthogonal group, SO(3), adapted to three-dimensional (3D) gait training and three-degree-of-freedom joint training is insufficient. The controller drives the patients to follow a timed trajectory and have the disadvantage of imposing a defined timing of movement. The deviation between patient and robot is neglected and the actual movement of human is not considered. Therefore, this study proposes a human-centred control method on SO(3) for AAN robotic rehabilitation. First, a feedback-stabilized closest attitude tracking algorithm is proposed to establish the angular velocity field and moment field on SO(3). And the framework of the field-based double-loop control is realised. The posture information of the human limbs is fed back to the controller through the sensing system, and the motion state of the human body is sensed in real-time, which is used as the basis for adjusting the torque to realise human-centred control. The experimental results show that different levels of adjustment torque are generated based on attitude deviations between the human and robot to verify the effectiveness of the designed controller.

GALOIS AUTOMORPHISMS AND CLASSICAL GROUPS

In a previous work, the second-named author gave a complete description of the action of automorphisms on the ordinary irreducible characters of the finite symplectic groups. We generalise this in two directions. Firstly, using work of the first-named author, we give a complete description of the action of Galois automorphisms on irreducible characters. Secondly, we extend both descriptions to cover the case of special orthogonal groups. As a consequence, we obtain explicit descriptions for the character fields of symplectic and special orthogonal groups.

Finding geodesics joining given points

Finding a geodesic joining two given points in a complete path-connected Riemannian manifold requires much more effort than determining a geodesic from initial data. This is because it is much harder to solve boundary value problems than initial value problems. Shooting methods attempt to solve boundary value problems by solving a sequence of initial value problems, and usually need a good initial guess to succeed. The present paper finds a geodesic gamma :[0,1]rightarrow M on the Riemannian manifold M with γ(0) = x0 and γ(1) = x1 by dividing the interval [0,1] into several sub-intervals, preferably just enough to enable a good initial guess for the boundary value problem on each subinterval. Then a geodesic joining consecutive endpoints (local junctions) is found by single shooting. Our algorithm then adjusts the junctions, either (1) by minimizing the total squared norm of the differences between associated geodesic velocities using Riemannian gradient descent, or (2) by solving a nonlinear system of equations using Newton’s method. Our algorithm is compared with the known leapfrog algorithm by numerical experiments on a 2-dimensional ellipsoid Ell(2) and on a left-invariant 3-dimensional special orthogonal group SO(3). We find Newton’s method (2) converges much faster than leapfrog when more junctions are needed, and that a good initial guess can be found for (2) by starting with Riemannian gradient descent method (1).

Compact Schur-Weyl duality and the Type B/C VW-algebra

In this paper, we study a compact analogue of Schur-Weyl duality for real orthogonal and symplectic groups. We analyse the functors Fμ,k defined in [12], and introduce a new algebra Bkθ. This algebra's finite dimensional modules are naturally a co-domain for the functors Fμ,k. With this new insight, we prove that principle series modules map to principles series modules of graded Hecke algebras and these functors preserve Langlands quotients and unitarity of Harish-Chandra modules.

Fredholm Pfaffian $$\tau $$-Functions for Orthogonal Isospectral and Isomonodromic Systems

We extend the approach to \(\tau \)-functions as Widom constants developed by Cafasso, Gavrylenko and Lisovyy to orthogonal loop group Drinfeld–Sokolov reductions and isomonodromic deformations systems. The combinatorial expansion of the \(\tau \)-function as a sum of correlators, each expressed as products of finite determinants, follows from using multicomponent fermionic vacuum expectation values of certain dressing operators encoding the initial conditions and dependence on the time parameters. When reduced to the orthogonal case, these correlators become finite Pfaffians and the determinantal \(\tau \)-functions, both in the Drinfeld–Sokolov and isomonodromic case, become squares of \(\tau \)-functions of Pfaffian type. The results are illustrated by several examples, consisting of polynomial \(\tau \)-functions of orthogonal Drinfeld–Sokolov type and isomonodromic ones with four regular singular points.

The Hilbert modular group and orthogonal groups

We derive an explicit isomorphism between the Hilbert modular group and certain congruence subgroups on the one hand and particular subgroups of the special orthogonal group SO(2, 2) on the other hand. The proof is based on an application of linear algebra adapted to number theoretical needs.

Traces on diagram algebras II: centralizer algebras of easy groups and new variations of the Young graph

In continuation of our recent work arXiv:2006.07312, we classify the extremal traces on infinite diagram algebras that appear in the context of Schur-Weyl duality for Banica and Speicher's easy groups. We show that the branching graphs of these algebras describe walks on new variations of the Young graph which describe curious ways of growing Young diagrams. As a consequence, we prove that the extremal traces on generic rook-Brauer algebras are always extensions of extremal traces on the group algebra $\mathbb{C}[S_{\infty}]$ of the infinite symmetric group. Moreover, we conjecture that the same is true for generic parameter deformations of the centralizers of the hyperoctahedral group and we reduce this conjecture to a conceptually much simpler numerical statement. Lastly, we address the trace classification problem for the Schur-Weyl dual of the halfliberated orthogonal group $O_N^*$, in which case extremal traces are always extensions of extremal traces on $\mathbb{C}[S_{\infty} \times S_{\infty}]$. Our approach relies on methods developed by Vershik and Nikitin.

An approach to p-adic qubits from irreducible representations of SO(3)p

We introduce the notion of a p-adic quantum bit (p-qubit) in the context of the p-adic quantum mechanics initiated and developed after the seminal paper of Volovich [Theor. Math. Phys. 71, 574 (1987)]. In this approach, physics takes place in a three-dimensional p-adic space rather than Euclidean space. Based on our prior work describing the p-adic special orthogonal group [Di Martino et al., arXiv:2104.06228 [math.NT] (2021)], we outline a program to classify its continuous unitary projective representations, which can be interpreted as a theory of p-adic angular momentum. The p-adic quantum bit arises from the irreducible representations of minimal nontrivial dimension two, of which we construct examples for all primes p.

The Quaternion Based Error Model Based on SE(3) of the INS

In this paper, an error model based on Special Euclidean group <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SE</i> (3) of inertial navigation system (INS) is proposed using the quaternion. The attitude and velocity are formulated together as element of the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SE</i> (3). The quaternion based error model has a better performance than the Euler Angles based error model, especially for the large initial attitude error. The conclusion can be also applied to the error model based on Lie group due to the superiority of the quaternion. Further, two field tests for the INS are carried out to investigate the benefits and drawbacks of the quaternion and Euler Angles based error model based on <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SE</i> (3) or Special orthogonal group <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SO</i> (3) for the INS.

Construction of local A-packets

Abstract In the present paper, we reformulate Mœglin’s explicit construction of local A-packets of split odd special orthogonal groups and symplectic groups. By this reformulation together with results of the previous paper with Mínguez, we can compute the A-packets explicitly. Also, we give a non-vanishing criterion of our parametrization, and an algorithm to compute certain derivatives. Finally, we prove a formula for the Aubert duals of irreducible representations of Arthur type.

Coordinate Descent on the Orthogonal Group for Recurrent Neural Network Training

We address the poor scalability of learning algorithms for orthogonal recurrent neural networks via the use of stochastic coordinate descent on the orthogonal group, leading to a cost per iteration that increases linearly with the number of recurrent states. This contrasts with the cubic dependency of typical feasible algorithms such as stochastic Riemannian gradient descent, which prohibits the use of big network architectures. Coordinate descent rotates successively two columns of the recurrent matrix. When the coordinate (i.e., indices of rotated columns) is selected uniformly at random at each iteration, we prove convergence of the algorithm under standard assumptions on the loss function, stepsize and minibatch noise. In addition, we numerically show that the Riemannian gradient has an approximately sparse structure. Leveraging this observation, we propose a variant of our proposed algorithm that relies on the Gauss-Southwell coordinate selection rule. Experiments on a benchmark recurrent neural network training problem show that the proposed approach is a very promising step towards the training of orthogonal recurrent neural networks with big architectures.

Multiplicity one theorems over positive characteristic

AbstractIn Aizenbud et al. (2010, Annals of Mathematics 172, 1407–1434), a multiplicity one theorem is proved for general linear groups, orthogonal groups, and unitary groups ( $GL, O,$ and U) over p-adic local fields. That is to say that when we have a pair of such groups $G_n{\subseteq } G_{n+1}$ , any restriction of an irreducible smooth representation of $G_{n+1}$ to $G_n$ is multiplicity-free. This property is already known for $GL$ over a local field of positive characteristic, and in this paper, we also give a proof for $O,U$ , and $SO$ over local fields of positive odd characteristic. These theorems are shown in Gan, Gross, and Prasad (2012, Sur les Conjectures de Gross et Prasad. I, Société Mathématique de France) to imply the uniqueness of Bessel models, and in Chen and Sun (2015, International Mathematics Research Notice 2015, 5849–5873) to imply the uniqueness of Rankin–Selberg models. We also prove simultaneously the uniqueness of Fourier–Jacobi models, following the outlines of the proof in Sun (2012, American Journal of Mathematics 134, 1655–1678).By the Gelfand–Kazhdan criterion, the multiplicity one property for a pair $H\leq G$ follows from the statement that any distribution on G invariant to conjugations by H is also invariant to some anti-involution of G preserving H. This statement for $GL, O$ , and U over p-adic local fields is proved in Aizenbud et al. (2010, Annals of Mathematics 172, 1407–1434). An adaptation of the proof for $GL$ that works over of local fields of positive odd characteristic is given in Mezer (2020, Mathematische Zeitschrift 297, 1383–1396). In this paper, we give similar adaptations of the proofs of the theorems on orthogonal and unitary groups, as well as similar theorems for special orthogonal groups and for symplectic groups. Our methods are a synergy of the methods used over characteristic 0 (Aizenbud et al. [2010, Annals of Mathematics 172, 1407–1434]; Sun [2012, American Journal of Mathematics 134, 1655–1678]; and Waldspurger [2012, Astérisque 346, 313–318]) and of those used in Mezer (2020, Mathematische Zeitschrift 297, 1383–1396).

Symmetric Divergence-free tensors in the Calculus of Variations

Divergence-free symmetric tensors seem ubiquitous in Mathematical Physics. We show that this structure occurs in models that are described by the so-called variational principle, where the argument of the Lagrangian is a closed differential form. Divergence-free tensors are nothing but the second form of the Euler--Lagrange equations. The symmetry is associated with the invariance of the Lagrangian density upon the action of some orthogonal group.

Noncommutative Furstenberg boundary

We introduce and study the notions of boundary actions and of the Furstenberg boundary of a discrete quantum group. As for classical groups, properties of boundary actions turn out to encode significant properties of the operator algebras associated with the discrete quantum group in question; for example we prove that if the action on the Furstenberg boundary is faithful, the quantum group C*-algebra admits at most one KMS-state for the scaling automorphism group. To obtain these results we develop a version of Hamana's theory of injective envelopes for quantum group actions, and establish several facts on relative amenability for quantum subgroups. We then show that the Gromov boundary actions of free orthogonal quantum groups, as studied by Vaes and Vergnioux, are also boundary actions in our sense; we obtain this by proving that these actions admit unique stationary states. Moreover, we prove these actions are faithful, hence conclude a new unique KMS-state property in the general case, and a new proof of unique trace property when restricted to the unimodular case. We prove equivalence of simplicity of the crossed products of all boundary actions of a given discrete quantum group, and use it to obtain a new simplicity result for the crossed product of the Gromov boundary actions of free orthogonal quantum groups.

Robust attitude tracking on the Special Orthogonal Group [formula omitted] using PD-type state feedback and linear matrix inequalities

The problem of rigid-body attitude tracking in the presence of exogenous disturbances is addressed. Attitude is parameterized using the rotation matrix, an element of the Special Orthogonal Group SO(3), as it provides a singularity-free and unambiguous attitude description. The closed-loop stability and robustness properties of a PD-type state-feedback control law, proposed in literature for attitude tracking using rotation matrices, are investigated using the nonlinear H∞ control framework. Starting from a dissipation inequality, sufficient conditions are derived which ensure that the closed-loop energy gain from bounded, finite-energy exogenous disturbances to a specified error signal respects a given upper bound. Then, the sufficient conditions are reformulated using the state and input matrices for the translational double integrator, and recast as linear matrix inequalities (LMIs). Lastly, the reformulated LMIs are used to synthesize controller gains for the proportional and derivative state-feedback terms in the original SO(3) control law. The controller synthesis problem for a microsatellite is considered as a case study. The controller gains are obtained using the proposed LMI-based procedure, and the tracking and disturbance rejection capabilities of the SO(3) controller are illustrated.