Brownian motion and the classical groups

Classical groups over finite fields, including symplectic groups, unitary groups and orthogonal groups, are finite simple groups of crucial importance. The structure and order of classical groups are the basic problems in group theory. Geometry of classical groups has been found extremely useful in many fields, such as combinatorial mathematics, coding and cryptographic theory. In this paper, a class of quasi-classical groups are introduced, which partially generalize the concepts of symplectic groups and orthogonal groups. We determine the structure and order of quasi-classical groups. As an application, we discuss the relationship between quasi-classical groups and self-dual normal bases over finite fields.

In this paper we deduce a graded version of Quillen–Suslin’s Local-Global Principle for the traditional classical groups, viz. general linear, symplectic and orthogonal groups and establish its equivalence of the normality property of the respective elementary subgroups. This generalizes previous result of Basu–Rao–Khanna. Then, as an application, we establish an analogue Local-Global Principle for the commutator subgroups of the special linear and symplectic groups. Finally, by using Swan–Weibel’s homotopy trick, we establish graded analogue of the Local-Global Principle for the transvection subgroups of the full automorphism groups of the linear, symplectic and orthogonal modules. This generalizes the previous result of Bak–Basu–Rao.

The classification of irreducible, spherical characters of the infinite-dimensional unitary/orthogonal/symplectic groups can be obtained by finding all possible limits of normalized, irreducible characters of the corresponding finite-dimensional groups, as the rank tends to infinity. We solve a q-deformed version of the latter problem for orthogonal and symplectic groups, extending previously known results for the unitary group. The proof is based on novel determinantal and double-contour integral formulas for the q-specialized characters.

By the methods used heretofore for the determination of the automorphisms of certain families of linear groups, for example, the (projective) unimodular, orthogonal, symplectic, and unitary groups (7, 8), it has been necessary to consider the various families separately and to give many case-by-case discussions, especially when the underlying vector space has few elements, even though the final results are very much the same for all of the groups. The purpose of this article is to give a completely uniform treatment of this problem for all the known finite simple linear groups (listed in §2 below). Besides the * ‘classical groups” mentioned above, these include the “exceptional groups,” considered over the complex field by Cartan and over an arbitrary field by Dickson, Chevalley, Hertzig, and the author (3, 4, 5, 6, 10, 15). The automorphisms of the latter groups are given here for the first time.

We classify the irreducible complex characters of the symplectic groups Sp 2n (q) and the orthogonal groups , Spin 2n+1(q) of degrees up to the bound D, where D = (q n − 1)q 4n−10/2 for symplectic groups, D = q 4n−8 for orthogonal groups in odd dimension, and D = q 4n−10 for orthogonal groups in even dimension.

A supercharacter theory for the Sylow p-subgroups of the finite symplectic and orthogonal groups

Chern–Simons theory, 2d Yang–Mills, and Lie algebra wanderers

We introduce a Lohe group which is a new class of matrix Lie groups and present a continuous dynamical system for the synchronization of group elements in a Lohe group. The Lohe group includes classical Lie groups such as the orthogonal, unitary, and symplectic groups, and since Lohe groups need not be compact, global existence of ODEs may fail. The proposed dynamical system generalizes the Lohe model (Lohe in J Phys A 43:465301, 2010; Lohe in J Phys A 42:395101–395126, 2009) itself a nonabelian generalization of the Kuramoto model, and alongside we also generalize the analytical framework (Ha and Ryoo in J Stat Phys 163:411–439, 2016) of emergent and unique phase-locked states. For the construction of the phase-locked states, we introduce Lyapunov functions measuring the ensemble diameter and the dissimilarity between two Lohe flows, and derive Gronwall-type differential inequalities for them. The global existence of solutions then become a consequence of the boundedness of these Lyapunov functions. Our sufficient framework for the emergent dynamics is formulated in terms of coupling strength and initial states, and it leads to the global existence of solutions and the formation and uniqueness of a phase-locked asymptotic state. As a concrete example, we demonstrate how our theory can show emergent phenomenon on the Heisenberg group, where all initial configurations tend to a unique phase-locked state exponentially fast.

We are concerned with finite-dimensional classical groups over arbitrary commutative fields. In an orthogonal group a Siegel transvection, that is, an element centralizing l⊥ for some totally singular 2-dimensional subspace l, plays the same role as a transvection in the linear, symplectic or unitary groups. The Main Theorem of this paper describes the possible embeddings of classical groups in classical groups such that (Siegel) transvections act as (Siegel) transvections.

In 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).

Bulk Witten indices and the number of normalizable ground states in supersymmetric quantum mechanics of orthogonal, symplectic and exceptional groups

In this paper we prove isomorphisms between 5 Lie groups (of arbitrary dimension and fixed signatures) in Clifford algebra and classical matrix Lie groups - symplectic, orthogonal and linear groups. Also we obtain isomorphisms of corresponding Lie algebras.

Optimization problems over compact Lie groups have been extensively studied due to their broad applications in linear programming and optimal control. This paper analyzes least square problems over a noncompact Lie group, the symplectic group $\Sp(2N,\R)$, which can be used to assess the optimality of control over dynamical transformations in classical mechanics and quantum optics. The critical topology for minimizing the Frobenius distance from a target symplectic transformation is solved. It is shown that the critical points include a unique local minimum and a number of saddle points. The topology is more complicated than those of previously studied problems on compact Lie groups such as the orthogonal and unitary groups because the incompatibility of the Frobenius norm with the pseudo-Riemannian structure on the symplectic group brings significant nonlinearity to the problem. Nonetheless, the lack of traps guarantees the global convergence of local optimization algorithms.

The list of known finite simple groups other than the cyclic, alternating, and Mathieu groups consists of the classical groups which are (projective) unimodular, orthogonal, symplectic, and unitary groups, the exceptional groups which are the direct analogues of the exceptional Lie groups, and certain twisted types which are constructed with the aid of Lie theory (see §§3 and 4 below). In this article, it is proved that each of these groups is generated by two of its elements. It is possible that one of the generators can be chosen of order 2, as is the case for the projective unimodular group (1), or even that one of the generators can be chosen as an arbitrary element other than the identity, as is the case for the alternating groups. Either of these results, if true, would quite likely require methods much more detailed than those used here.

Maximal Overgroups of Singer Elements in Classical Groups