Symplectic Group Research Articles

On Spectra of Almost Simple Extensions of Even-Dimensional Orthogonal Groups

The spectrum of a finite group is the set of the orders of its elements. We consider the problem that arises within the framework of recognition of finite simple groups by spectrum: Determine all finite almost simple groups having the same spectrum as its socle. This problem was solved for all almost simple groups with exception of the case that the socle is a simple even-dimensional orthogonal group over a field of odd characteristic. Here we address this remaining case and determine the almost simple groups in question. Also we prove that there are infinitely many pairwise nonisomorphic finite groups having the same spectrum as the simple 8-dimensional symplectic group over a field of characteristic other than 7.

Characterization of Simple Symplectic Groups of Degree 4 over Locally Finite Fields in the Class of Periodic Groups

Let G be a periodic group containing an element of order 2 such that each of its finite subgroups of even order lies in a finite subgroup isomorphic to a simple symplectic group of degree 4. It is shown that G is isomorphic to a simple symplectic group S4(Q) of degree 4 over some locally finite field Q.

Hyperquaternions: A New Tool for Physics

A hyperquaternion formulation of Clifford algebras in n dimensions is presented. The hyperquaternion algebra is defined as a tensor product of quaternion algebras $$\mathbb {H}$$ (or a subalgebra thereof). An advantage of this formulation is that the hyperquaternion product is defined independently of the choice of the generators. The paper gives an explicit expression of the generators and develops a generalized multivector calculus. Due to the isomorphism $$\mathbb {H}\otimes \mathbb {H}\simeq m(4, \mathbb {R})$$ , hyperquaternions yield all real, complex and quaternion square matrices. A hyperconjugation is introduced which generalizes the concepts of transposition, adjunction and transpose quaternion conjugate. As applications, simple expressions of the unitary and unitary symplectic groups are obtained. Finally, the hyperquaternions are compared, in the context of physical applications, to another algebraic structure based on octonions which has been proposed recently.

Appendix: Non-uniqueness of supercuspidal support for finite reductive groups

We exhibit a simple representation of the finite symplectic group of Sp8(q) in characteristic ℓ|q2+1 having two non-conjugate supercuspidal supports. This answers a question of M.-F. Vignéras.

Odd orthogonal matrices and the non-injectivity of the Vaserstein symbol

R.A. Rao–W. van der Kallen showed that the Vaserstein symbol VΓ(SR3) from the orbit space of unimodular rows of length three over the coordinate ring of the real three sphere SR3 modulo elementary action to the elementary symplectic Witt group WE(Γ(SR3)) is not injective. Dhvanita R. Rao–Neena Gupta gave an uncountable family of singular real threefolds Aα for which the Vaserstein symbol VAα is not injective. In this paper, we give a countable family of smooth real birationally equivalent threefolds An for which the Vaserstein symbol VAn is not injective.

Normalizers of Elementary Overgroups of Ep(2, A)

Let A be an involution ring, e1 , . . . , en be a full system of Hermitian idempotents in A, let every ei generate A as a two-sided ideal, and 2 ∈ A∗. In this paper, the normalizers of the groups Ep(2,A) · E(2,A, I) are calculated under natural assumptions on A, where Ep(2,A) denotes the elementary symplectic group, E(2,A, I) stands for the elementary subgroup of level I.

Representation stability for filtrations of Torelli groups

We show, finitely generated rational $\mathsf{VIC}_{\mathbb Q}$-modules and $\mathsf{SI}_{\mathbb Q}$-modules are uniformly representation stable and all their submodules are finitely generated. We use this to prove two conjectures of Church and Farb, which state that the quotients of the lower central series of the Torelli subgroups of $\mathrm{Aut}(F_n)$ and $\mathrm{Mod}(\Sigma_{g,1})$ are uniformly representation stable as sequences of representations of the general linear groups and the symplectic groups, respectively. Furthermore we prove an analogous statement for their Johnson filtrations.

A Riemannian‐steepest‐descent approach for optimization on the real symplectic group

In this paper, we first give the geodesic in closed form on the real symplectic group endowed with a Riemannian metric and then study a geodesic‐based Riemannian‐steepest‐descent approach to compute the empirical average out of a set of symplectic matrices. The devised averaging algorithm is compared with the Euclidean gradient algorithm and the extended Hamiltonian algorithm. Simulation examples show that the convergence of the geodesic‐based Riemannian‐steepest‐descent algorithm is the fastest among the 3 considered algorithms.

Enumeration of Commuting Pairs in Lie Algebras over Finite Fields

Feit and Fine derived a generating function for the number of ordered pairs of commuting $$n{\times}n$$ matrices over the finite field $${\mathbb{F}_{q}}$$ . This has been reproved and studied by Bryan and Morrison from the viewpoint of motivic Donaldson-Thomas theory. In this note, we give a new proof of the Feit-Fine result, and generalize it to the Lie algebra of finite unitary groups and to the Lie algebra of odd characteristic finite symplectic groups. We extract some asymptotic information from these generating functions. Finally, we derive generating functions for the number of commuting nilpotent elements for the Lie algebras of the finite general linear and unitary groups, and of odd characteristic symplectic groups.

The Gromov width of coadjoint orbits of the symplectic group

We prove that the Gromov width of coadjoint orbits of the symplectic group is at least equal to the upper bound known from the works of Zoghi and Caviedes. This establishes the actual Gromov width. Our work relies on a toric degeneration of a coadjoint orbit to a toric variety. The polytope associated to this toric variety is a string polytope arising from a string parametrization of elements of a crystal basis for a certain representation of the symplectic group.

Generators and relations for the unitary group of a skew hermitian form over a local ring

Let (S,⁎) be an involutive local ring and let U(2m,S) be the unitary group associated to a nondegenerate skew hermitian form defined on a free S-module of rank 2m. A presentation of U(2m,S) is given in terms of Bruhat generators and their relations. This presentation is used to construct an explicit Weil representation of the symplectic group Sp(2m,R) when S=R is commutative and ⁎ is the identity.When S is commutative but ⁎ is arbitrary with fixed ring R, an elementary proof that the special unitary group SU(2m,S) is generated by unitary transvections is given. This is used to prove that the reduction homomorphisms SU(2m,S)→SU(2m,S˜) and U(2m,S)→U(2m,S˜) are surjective for any factor ring S˜ of S. The corresponding results for the symplectic group Sp(2m,R) are obtained as corollaries when ⁎ is the identity.

Real Representations of Finite Symplectic Groups over Fields of Characteristic Two

Abstract We prove that when q is a power of 2 every complex irreducible representation of $\textrm{Sp}\big (2n, \mathbb{F}_{q}\big )$ may be defined over the real numbers, that is, all Frobenius–Schur indicators are 1. We also obtain a generating function for the sum of the degrees of the unipotent characters of $\textrm{Sp}\big(2n, \mathbb{F}_{q}\big )$, or of $\textrm{SO}\big(2n+1,\mathbb{F}_{q}\big )$, for any prime power q.

On the Symplectic Structures in Frame Bundles and the Finite Dimension of Basic Symplectic Cohomologies

We present a construction (and classification) of certain invariant 2-forms on the real symplectic group. They are used to define a symplectic form on the quotient by a maximal torus and to "lift" a symplectic structure from a symplectic manifold to the bundle of frames. This is a by-product of a failed attempt to prove certain finiteness theorems for basic symplectic cohomologies. In the last part of the paper we include a valid proof.

Periods and (χ, b)-factors of cuspidal automorphic forms of symplectic groups

In this paper, we introduce a new family of period integrals attached to irreducible cuspidal automorphic representations $\sigma$ of symplectic groups $\mathrm{Sp}_{2n}(\mathbb{A})$, which detects the right-most pole of the $L$-function $L(s,\sigma\times\chi)$ for some character $\chi$ of $F^\times\backslash\mathbb{A}^\times$ of order at most $2$, and hence the occurrence of a simple global Arthur parameter $(\chi,b)$ in the global Arthur parameter $\psi$ attached to $\sigma$. We also give a characterisation of first occurrences of theta correspondence by (regularised) period integrals of residues of certain Eisenstein series.

Conjectures on correspondence of symplectic modular forms of middle parahoric type and Ihara lifts

By Ihara (J Math Soc Jpn 16:214–225, 1964) and Langlands (Lectures in modern analysis and applications III, lecture notes in math, vol 170. Springer, Berlin, pp 18–61, 1970), it is expected that automorphic forms of the symplectic group $$Sp(2,{\mathbb {R}})\subset GL_{4}({\mathbb {R}})$$ of rank 2 and those of its compact twist have a good correspondence preserving L functions. Aiming to give a neat classical isomorphism between automorphic forms of this type for concrete discrete subgroups like Eichler (J Reine Angew Math 195:156–171, 1955) and Shimizu (Ann Math 81(2):166–193, 1965) (and not aiming the general representation theory), in our previous papers Hashimoto and Ibukiyama (Adv Stud Pure Math 7:31–102, 1985) and Ibukiyama (J Fac Sci Univ Tokyo Sect IA Math 30:587–614, 1984; Adv Stud Pure Math 7:7–29 1985; in: Furusawa (ed) Proceedings of the 9-th autumn workshop on number theory, 2007), we have given two different conjectures on precise isomorphisms between Siegel cusp forms of degree two and automorphic forms of the symplectic compact twist USp(2), one is the case when subgroups of both groups are maximal locally, and the other is the case when subgroups of both groups are minimal parahoric. We could not give a good conjecture at that time when the discrete subgroups for Siegel cusp forms are middle parahoric locally (like $$\Gamma _0^{(2)}(p)$$ of degree two). Now a subject of this paper is a conjecture for such remaining cases. We propose this new conjecture with strong evidence of relations of dimensions and also with numerical examples. For the compact twist, it is known by Ihara that there exist liftings of Saito–Kurokawa type and of Yoshida type. It was not known about the description of the image of these liftings, but we can give here also a very precise conjecture on the image of the Ihara liftings.

Action of automorphisms on irreducible characters of symplectic groups

Assume G is a finite symplectic group Sp2n(q) over a finite field Fq of odd characteristic. We describe the action of the automorphism group Aut(G) on the set Irr(G) of ordinary irreducible characters of G. This description relies on the equivariance of Deligne–Lusztig induction with respect to automorphisms. We state a version of this equivariance which gives a precise way to compute the automorphism on the corresponding Levi subgroup; this may be of independent interest. As an application we prove that the global condition in Späth's criterion for the inductive McKay condition holds for the irreducible characters of Sp2n(q).

Symmetry breaking by bifundamentals

We derive all possible symmetry breaking patterns for all possible Higgs fields that can occur in intersecting brane models: bi-fundamentals and rank-2 tensors. This is a field-theoretic problem that was already partially solved in 1973 by Ling-Fong Li. In that paper the solution was given for rank-2 tensors of orthogonal and unitary group, and U(N x U(M) and O(N) x O(M) bi-fundamentals. We extend this first of al to symplectic groups. When formulated correctly, this turns out to be straightforward generalization of the previous results from real and complex numbers to quaternions. The extension to mixed bi-fundamentals is more challenging and interesting. The scalar potential has up to six real parameters. Its minima or saddle points are described by block-diagonal matrices built out of K blocks of size p x q. Here p=q=1 for the solutions of Ling-Fong Li, and the number of possibilities for p x q is equal to the number of real parameters in the potential, minus 1. The maximum block size is p x q=2 x 4. Different blocks cannot be combined, and the true minimum occurs for one choice of basic block, and for either K=1 or K maximal, depending on the parameter values.

Mirror theories of 3d mathcal{N} = 2 SQCD

Using a recently proposed duality for U(N ) supersymmetric QCD (SQCD) in three dimensions with monopole superpotential, in this paper we derive the mirror dual description of mathcal{N} = 2 SQCD with unitary gauge group, generalizing the known mirror dual description of abelian gauge theories. We match the chiral ring of the dual theories and their partition functions on the squashed sphere. We also conjecture a generalization for SQCD with orthogonal and symplectic gauge groups.

Congruence subgroups of braid groups

In this paper, we give a description of the generators of the prime level congruence subgroups of braid groups. Also, we give a new presentation of the symplectic group over a finite field, and we calculate the symmetric quotients of the prime level congruence subgroups of braid groups. Finally, we find a finite generating set for the level-3 congruence subgroup of the braid group on three strands.

The influence of order and conjugacy class length on the structure of finite groups

Let $2^{n}+1 \gt 5$ be a prime number. In this article, we will show $G\cong C_{n}(2)$ if and only if $|G|=|C_{n}(2)|$ and $G$ has a conjugacy class length ${|C_{n}(2)|}/({2^{n}+1})$. Furthermore, we will show Thompson's conjecture is valid under a weak condition for the symplectic groups $C_{n}(2)$.