Symplectic Group Research Articles

Lagrangian Grassmann manifold Λ(2)

Based on the relationship between symplectic group Sp(2) and Λ(2), we provide an intuitive explanation (model) of the 3-dimensional Lagrangian Grassmann manifold Λ(2), the singular cycles of Λ(2), and the special Lagrangian Grassmann manifold SΛ(2). Under this model, we give a formula of the rotation paths dened by Arnold.

A note on ‘gaugings’ in four spacetime dimensions and electric-magnetic duality

The variety of consistent ‘gauging’ deformations of supergravity theories in four dimensions depends on the choice of Lagrangian formulation. One important goal is to get the most general deformations without making hidden assumptions. Ignoring supersymmetry we consider in this paper nv abelian vector potentials in four spacetime dimensions with non-minimal kinetic coupling to ns uncharged (possibly nonlinear) scalar fields. As in the case of extended supergravities, one model may possess different formulations related by . The symplectic group mixes its electric and magnetic potentials. The model admits a global duality symmetry subgroup G which acts also on the scalars. We recall first how the general second order Lagrangian, its local deformations and those of its abelian gauge group will depend on the choice of 2nv directions (choice of ‘Darboux frame’). We start from a general frame defined by the symplectic transformation relating it to a fixed ‘reference’ one. Combinations of symplectic matrix coefficients appear then as constant parameters in the second order Lagrangians. Another gauging method uses an ‘embedding tensor’ that characterizes the realization of the gauge group via the global duality group. It involves additional 2-form gauge fields. A suitable zero charge limit of this realization has abelian gauge group and the ‘gauging’ can be viewed as a consistent deformation of that limit. We show that the two methods applied to the corresponding ungauged models have equivalent local deformations—and more generally, have isomorphic local BRST cohomology at all ghost numbers. We finally consider manifestly duality invariant first order actions with abelian gauge group. We point out that obstructions to non-abelian deformations of the Yang–Mills type exhibited in a previous work remain present when couplings to scalar fields are included.

Simultaneous computation of Hecke operators

We present a new method for computing Hecke operators acting on spaces of algebraic modular forms based on an idea of Eichler's. The method significantly improves the complexity of computing operators that can be obtained this way and in addition computes two operators simultaneously. We show that in certain cases the approach can be used to attain the action of the full Hecke algebra with respect to a hyperspecial subgroup and use it to compute Hecke eigenforms for compact forms of symplectic groups.

On the second fundamental theorem of invariant theory for the orthosymplectic supergroup

Let OSp(V) be the orthosymplectic supergroup on an orthosymplectic vector superspace V of superdimension (m|2n). Lehrer and Zhang showed that there is a surjective algebra homomorphism Frr:Br(m−2n)→EndOSp(V)(V⊗r), where Br(m−2n) is the Brauer algebra of degree r with parameter m−2n. The second fundamental theorem of invariant theory in this setting seeks to describe the kernel KerFrr of Frr as a 2-sided ideal of Br(m−2n). In this paper, we show that KerFrr≠0 if and only if r≥rc:=(m+1)(n+1), and give a basis and a dimension formula for KerFrr. We show that KerFrr as a 2-sided ideal of Br(m−2n) is generated by KerFrcrc for any r≥rc, and we provide an explicit set of generators for KerFrcrc. These generators coincide in the classical case with those obtained in recent papers of Lehrer and Zhang on the second fundamental theorem of invariant theory for the orthogonal and symplectic groups. As an application we obtain the necessary and sufficient conditions for the endomorphism algebra Endosp(V)(V⊗r) over the orthosymplectic Lie superalgebra osp(V) to be isomorphic to Br(m−2n).

Double groupoids and the symplectic category

We introduce the notion of a symplectic hopfoid, a 'groupoid-like' object in the category of symplectic manifolds whose morphisms are given by canonical relations. Such groupoid-like objects arise when applying a version of the cotangent functor to the structure maps of a Lie groupoid. We show that such objects are in one-to-one correspondence with symplectic double groupoids, generalizing a result of Zakrzewski concerning symplectic double groups and Hopf algebra objects in the aforementioned category. The proof relies on a new realization of the core of a symplectic double groupoid as a symplectic quotient of the total space. The resulting constructions apply more generally to give a correspondence between double Lie groupoids and groupoid-like objects in the category of smooth manifolds and smooth relations, and we show that the cotangent functor relates the two constructions.

THE GEOMETRY OF BLUEPRINTS PART II: TITS–WEYL MODELS OF ALGEBRAIC GROUPS

This paper is dedicated to a problem raised by Jacquet Tits in 1956: the Weyl group of a Chevalley group should find an interpretation as a group over what is nowadays called$\mathbb{F}_{1}$,the field with one element. Based on Part I of The geometry of blueprints, we introduce the class ofTits morphismsbetween blue schemes. The resultingTits category$\text{Sch}_{{\mathcal{T}}}$comes together with a base extension to (semiring) schemes and the so-calledWeyl extensionto sets. We prove for${\mathcal{G}}$in a wide class of Chevalley groups—which includes the special and general linear groups, symplectic and special orthogonal groups, and all types of adjoint groups—that a linear representation of${\mathcal{G}}$defines a model$G$in$\text{Sch}_{{\mathcal{T}}}$whose Weyl extension is the Weyl group$W$of${\mathcal{G}}$. We call such modelsTits–Weyl models. The potential of Tits–Weyl models lies in(a)their intrinsic definition that is given by a linear representation;(b)the (yet to be formulated) unified approach towards thick and thin geometries; and(c)the extension of a Chevalley group to a functor on blueprints, which makes it, in particular, possible to consider Chevalley groups over semirings. This opens applications to idempotent analysis and tropical geometry.

A Local-Global Principle for Symplectic $\mathrm K_2$

We prove that an element of the relative symplectic Steinberg group g\in\mathrm{StSp}_{2n}(R[t],\,tR[t]) is trivial if and only if its image under any maximal localisation homomorphism is trivial.

The number of independent Traces and Supertraces on the Symplectic Reflection Algebra H1,η(Γ≀SN)

Symplectic reflection algebra H1η(G) has a T(G)-dimensional space of traces whereas, when considered as a superalgebra with a natural parity, it has an S(G)-dimensional space of supertraces. The values of T(G) and S(G) depend on the symplectic reflection group G and do not depend on the parameter η.In this paper, the values T(G) and S(G) are explicitly calculated for the groups G = Γ≀SN, where Γ is a finite subgroup of Sp(2,ℂ).

On the Symplectic Covariance and Interferences of Time-Frequency Distributions

We study the covariance property of quadratic time-frequency distributions with respect to the action of the extended symplectic group. We show how covariance is related to and in fact in competition with the possibility of damping the interferences which arise due to the quadratic nature of the distributions. We also show that the well-known fully covariance property of the Wigner distribution in fact characterizes it (up to a constant factor) among the quadratic distributions $L^{2}(\mathbb{R}^{n})\rightarrow C_{0}({ \mathbb{R}^{2n}})$. A similar characterization for the closely related Weyl transform is given as well. The results are illustrated by several numerical experiments for the Wigner and Born--Jordan distributions of the sum of four Gaussian functions in the so-called diamond configuration.

Differential and Integral Projective Invariants for the Groups of Diffeomorphisms

In this paper we study the differential and integral invariants for the action of the projective group PGL(n + 1) on the group of diffeomorphisms Diff(ℝP n ) by conjugations. Cases n = 1 and n = 2 are considered. For n = 1 the algebra of differential invariants is found and the criterion of the local equivalence of two diffeomorphisms is obtained. Also several integral invariants for n = 1 and n = 2 are calculated, the analogy with Calaby integral invariant for the symplectic groups is established.

Character expansion of matrix integrals

We consider expansions of certain multiple integrals and BKP tau functions in characters of orthogonal and symplectic groups. In particular we consider character expansions of integrals over orthogonal and over symplectic matrices.

The E-normal structure of odd dimensional unitary groups

In this paper we define odd dimensional unitary groups U2n+1(R,Δ). These groups contain as special cases the odd dimensional general linear groups GL2n+1(R) where R is any ring, the odd dimensional orthogonal and symplectic groups O2n+1(R) and Sp2n+1(R) where R is any commutative ring and further the first author's even dimensional unitary groups U2n(R,Λ) where (R,Λ) is any form ring. We classify the E-normal subgroups of the groups U2n+1(R,Δ) (i.e. the subgroups which are normalized by the elementary subgroup EU2n+1(R,Δ)), under the condition that R is either a semilocal or quasifinite ring with involution and n≥3. Further we investigate the action of U2n+1(R,Δ) by conjugation on the set of all E-normal subgroups.

On intersective polynomials with non-solvable Galois group

ABSTRACTWe present new theoretical results on the existence of intersective polynomials (that is, integer polynomials with roots in all ℚp, but not in ℚ) with certain prescribed Galois groups, namely the projective and affine linear groups PGL2(ℓ) and AGL2(ℓ) as well as the affine symplectic groups . For further families of affine groups, existence results are proven conditional on the existence on certain tamely ramified Galois extensions. We also compute explicit families of intersective polynomials for certain non-solvable groups.

Book Review: The endoscopic classification of representations—orthogonal and symplectic groups

Book Review: The endoscopic classification of representations—orthogonal and symplectic groups

On locally conformal symplectic manifolds of the first kind

We present some examples of locally conformal symplectic structures of the first kind on compact nilmanifolds which do not admit Vaisman metrics. One of these examples does not admit locally conformal Kähler metrics and all the structures come from left-invariant locally conformal symplectic structures on the corresponding nilpotent Lie groups. Under certain topological restrictions related with the compactness of the canonical foliation, we prove a structure theorem for locally conformal symplectic manifolds of the first kind. In the non compact case, we show that they are the product of a real line with a compact contact manifold and, in the compact case, we obtain that they are mapping tori of compact contact manifolds by strict contactomorphisms. Motivated by the aforementioned examples, we also study left-invariant locally conformal symplectic structures on Lie groups. In particular, we obtain a complete description of these structures (with non-zero Lee form) on connected simply connected nilpotent Lie groups in terms of locally conformal symplectic extensions and symplectic double extensions of symplectic nilpotent Lie groups. In order to obtain this description, we study locally conformal symplectic structures of the first kind on Lie algebras.

Fischer decomposition for the symplectic group

We prove the Fischer decomposition for the space of spinor-valued polynomials, defined on Euclidean space of four-fold dimension, in terms of irreducible modules for the symplectic group, consisting of so-called osp(4|2)-monogenics.

Jacquet–Langlands–Shimizu correspondence for theta lifts to $GSp(2)$ and its inner forms I: An explicit functorial correspondence

As was first essentially pointed out by Tomoyoshi Ibukiyama, Hecke eigenforms on the indefinite symplectic group $GSp(1,1)$ or the definite symplectic group $GSp^*(2)$ over $\mathbb{Q}$ right invariant by a (global) maximal open compact subgroup are conjectured to have the same spinor $L$-functions as those of paramodular new forms of some specified level on the symplectic group $GSp(2)$ (or $GSp(4)$). This can be viewed as a generalization of the Jacquet–Langlands–Shimizu correspondence to the case of $GSp(2)$ and its inner forms $GSp(1,1)$ and $GSp^*(2)$. In this paper we provide evidence of the conjecture on this explicit functorial correspondence with theta lifts: a theta lift from $GL(2)\times B^{\times}$ to $GSp(1,1)$ or $GSp^*(2)$ and a theta lift from $GL(2)\times GL(2)$ (or $GO(2,2)$) to $GSp(2)$. Here $B$ denotes a definite quaternion algebra over $\mathbb{Q}$. Our explicit functorial correspondence given by these theta lifts are proved to be compatible with archimedean and non-archimedean local Jacquet–Langlands correspondences. Regarding the non-archimedean local theory we need some explicit functorial correspondence for spherical representations of the inner form and non-supercuspidal representations of $GSp(2)$, which is studied in the appendix by Ralf Schmidt.

Degenerations of Kählerian K3 surfaces with finite symplectic automorphism groups. III

Following our papers [1] and [2] (Parts I and II), we classify degenerations of codimension 2 or more of Kählerian surfaces with finite symplectic automorphism groups. In [1] and [2] this was done for codimension .

Lattices generated by joins of the flats in orbits under finite affine-singular symplectic group and its characteristic polynomials

Let ASG(2ν + l, ν;F q ) be the (2ν + l)-dimensional affine-singular symplectic space over the finite field F q and ASp2ν+l,ν (F q ) be the affine-singular symplectic group of degree 2ν + l over F q . Let O be any orbit of flats under ASp2ν+l,ν (F q ). Denote by L J the set of all flats which are joins of flats in O such that O ⊆ L J and assume the join of the empty set of flats in ASG(2ν + l, ν;F q ) is ∅. Ordering L J by ordinary or reverse inclusion, then two lattices are obtained. This paper firstly studies the inclusion relations between different lattices, then determines a characterization of flats contained in a given lattice L J , when the lattices form geometric lattice, lastly gives the characteristic polynomial of L J .

Vanishing of L2–Betti numbers and failure of acylindrical hyperbolicity of matrix groups over rings

Let $R$ be an infinite commutative ring with identity and $n\geq 2$ be an integer. We prove that for each integer $i=0,1,\cdots ,n-2,$ the $L^{2}$-Betti number $b_{i}^{(2)}(G)=0,$ $\ $when $G=\mathrm{GL}_{n}(R)$ the general linear group, $\mathrm{SL}_{n}(R)$ the special linear group, $% E_{n}(R)$ the group generated by elementary matrices. When $R$ is an infinite principal ideal domain, similar results are obtained for $\mathrm{Sp}_{2n}(R)$ the symplectic group, $\mathrm{ESp}_{2n}(R)$ the elementary symplectic group, $\mathrm{O}(n,n)(R)$ the split orthogonal group or $\mathrm{EO}(n,n)(R)$ the elementary orthogonal group. Furthermore, we prove that $G$ is not acylindrically hyperbolic if $n\geq 4$. We also prove similar results for a class of noncommutative rings. The proofs are based on a notion of $n$-rigid rings.