Symplectic Group Sp Research Articles

Schur rings over Sp(n,2) and multiplicity one subgroups

We study commutative Schur rings over the symplectic groups Sp(n,2) that contain the class C of symplectic transvections. We find the possible partitions of C determined by the Schur ring. We show how this restricts the possibilities for multiplicity one subgroups of Sp(n,2).

Joint moments of derivatives of characteristic polynomials of random symplectic and orthogonal matrices

We investigate the joint moments of derivatives of characteristic polynomials over the unitary symplectic group Sp(2N) and the orthogonal ensembles SO(2N) and O−(2N) . We prove asymptotic formulae for the joint moments of the n 1th and n 2th derivatives of the characteristic polynomials for all three matrix ensembles. Our results give two explicit formulae for each of the leading order coefficients, one in terms of determinants of hypergeometric functions and the other as combinatorial sums over partitions. We use our results to put forward conjectures on the joint moments of derivatives of L-functions with symplectic and orthogonal symmetry.

Weakly interacting Bose gas with two-body losses

We study the many-body dynamics of weakly interacting Bose gases with two-particle losses. We show that both the two-body interactions and losses in atomic gases may be tuned by controlling the inelastic scattering process between atoms by an optical Feshbach resonance. Interestingly, the low-energy behavior of the scattering amplitude is governed by a single parameter, i.e. the complex ss-wave scattering length a_cac. The many-body dynamics are thus described by a Lindblad master equation with complex scattering length. We solve this equation by applying the Bogoliubov approximation in analogy to the closed systems. Various peculiar dynamical properties are discovered, some of them may be regarded as the dissipative counterparts of the celebrated results in closed Bose gases. For example, we show that the next-order correction to the mean-field particle decay rate is to the order of |n a_c^{3}|^{1/2}|nac3|1/2, which is an analogy of the Lee-Huang-Yang correction of Bose gases. It is also found that there exists a dynamical symmetry of symplectic group Sp(4,\mathbb{C})(4,ℂ) in the quadratic Bogoliubov master equation, which is an analogy of the SU(1,1) dynamical symmetry of the corresponding closed system. We further confirmed the validity of the Bogoliubov approximation by comparing its results with a full numerical calculation in a double-well toy model. Generalizations of other alternative approaches such as the dissipative version of the Gross-Pitaevskii equation and hydrodynamic theory are also discussed in the last.

Solving the matrix exponential function for the Lie groups SU(3), SU(4) and Sp(2)

The well known analytical formula for SU(2) matrices U = exp (i vec tau !cdot ! vec varphi ,) = cos |vec varphi ,|mathbf{1} + ivec tau !cdot ! {hat{varphi }} , sin |vec varphi ,| is extended to the SU(3) group with eight real parameters. The resulting analytical formula involves the sum over three real roots of a cubic equation, corresponding to the so-called irreducible case, where one has to employ for solution the trisection of an angle. When going to the special unitary group SU(4) with 15 real parameters, the analytical formula involves the sum over four real roots of a quartic equation. The associated cubic resolvent equation with three positive roots belongs again to the irreducible case. Furthermore, by imposing the pertinent condition on SU(4) matrices one can also treat the symplectic group Sp(2) with ten real parameters. Since there the roots occur as two pairs of opposite sign, this simplifies the analytical formula for Sp(2) matrices considerably. An outlook to the situation with quasi-analytical formulas for SU(5), SU(6) and Sp(3) is also given.

Auto-correlation functions of Sato–Tate distributions and identities of symplectic characters

The Sato–Tate distributions for genus 2 curves (conjecturally) describe the statistics of numbers of rational points on the curves. In this paper, we explicitly compute the auto-correlation functions of Sato–Tate distributions for genus 2 curves as sums of irreducible characters of symplectic groups. Our computations bring about families of identities involving irreducible characters of symplectic groups Sp(2m) for all m∈Z≥1, which have interest in their own rights.

A remarkable dynamical symmetry of the Landau problem

We show that the dynamical group of an electron in a constant magnetic feld is the group of symplectomorphisms Sp(4, R). It is generated by the spinorial realization of the conformal algebra so(2,3) considered in Dirac’s seminal paper ”A Remarkable Representation of the 3 + 2 de Sitter Group”. The symplectic group Sp(4,R) is the double covering of the conformal group SO(2,3) of 2+1 dimensional Minkowski spacetime which is in turn the dynamical group of a hydrogen atom in 2 space dimensions. The Newton-Hooke duality between the 2D hydrogen atom and the Landau problem is explained via the Tits-Kantor-Koecher construction of the conformal symmetries of the Jordan algebra of real symmetric 2 × 2 matrices. The connection between the Landau problem and the 3D hydrogen atom is elucidated by the reduction of a Dirac spinor to a Majorana one in the Kustaanheimo-Stiefel spinorial regularization.

Harmonic analysis of boxed hyperoctahedral Hall-Littlewood polynomials

For positive integers n and c with c≥2n (= the Coxeter number of the hyperoctahedral group of signed permutations of degree n), we present a finite-dimensional discrete orthogonality relation for Macdonald's three-parameter hyperoctahedral Hall-Littlewood polynomials of degree at most c in each of the n variables. These polynomials are labeled by partitions λ that fit inside a rectangular box of shape cn, i.e. the partitions in question are of length ≤n and have parts of size ≤c. We employ coordinate patches around the vertices of the alcove of boxed partitions Λ(n,c)={λ⊆cn} to establish the self-adjointness of a one-parameter family of commuting discrete difference operators ▪ acting on functions f:Λ(n,c)→C. By construction, the basis of hyperoctahedral Hall-Littlewood polynomials constitutes a joint eigenbasis for ▪ with simple spectrum, which gives rise to our discrete orthogonality relation. From the point of view of quantum integrable particle dynamics, the present geometric construction establishes the orthogonality of the Bethe Ansatz eigenfunctions for a recently studied q-boson system on a finite lattice with integrable open-end boundary conditions. The Bethe Ansatz equations enter in the geometric picture as compatibility conditions between coordinate patches stemming from distinct vertices. Two applications of the orthogonality relations are highlighted: (i) a cubature rule for the integration of symmetric functions with respect to the Haar measure on the compact symplectic group Sp(n), and (ii) a Verlinde formula for the structure constants of a deformation of the Wess-Zumino-Witten fusion ring associated with the affine Lie algebra of type Cˆn (= Cn(1)).

Study of Abelian dominance in Sp(N) QCD in the de Sitter background

We propose the effective theory based on the compact symplectic group Sp(N) in the new quadratic gauge on the 4-dimensional de Sitter spacetime (dS 4) with a view to investigate the infrared regime in it. Abelian dominance is characterized by off-diagonal gluons acquiring dynamical masses and it hints at existence of confinement. We do observe Abelian dominance in the ghost condensed vacuum of this theory in the de Sitter background. Along with this observation, we find an unusual result that the mass of an off-diagonal gluon on dS 4 is position dependent as a consequence of the curved geometry. In our previous papers, we have shown that Abelian dominance is strongly indicated in the Minkowski spacetime too. Therefore, the present observation suggests that the curvature of the dS 4 does not change the infrared behaviour of the present theory on dS 4 from that of the quadratic gauge fixed theory in Minkowski spacetime, but it causes the mass of an off-diagonal gluon on the dS 4 to be position dependent.

Invariant metric on the extended Siegel–Jacobi upper half space

The real Jacobi group GnJ(R), defined as the semidirect product of the Heisenberg group Hn(R) with the symplectic group Sp(n,R), admits a matrix embedding in Sp(n+1,R). The modified pre-Iwasawa decomposition of Sp(n,R) allows us to introduce a convenient coordinatization Sn of GnJ(R), which for G1J(R) coincides with the S-coordinates. Invariant one-forms on GnJ(R) are determined. The formula of the 4-parameter invariant metric on G1J(R) obtained as sum of squares of 6 invariant one-forms is extended to GnJ(R), n∈N. We obtain a three parameter invariant metric on the extended Siegel–Jacobi upper half space X̃nJ≈XnJ×R by adding the square of an invariant one-form to the two-parameter balanced metric on the Siegel–Jacobi upper half space XnJ=GnJ(R)U(n)×R.

Fibred algebraic surfaces and commutators in the Symplectic group

We describe the minimal number of critical points and the minimal number s of singular fibres for a non isotrivial fibration of a surface S over a curve B of genus 1, exhibiting several examples and in particular constructing a fibration with s=1 and irreducible singular fibre with 4 nodes.Then we consider the associated factorizations in the mapping class group and in the symplectic group. We describe explicitly which products of transvections on homologically independent and disjoint circles are a commutator in the Symplectic group Sp(2g,Z).

Characters of some unitary highest weight representations via the theta correspondence

In this article, we consider a dual pair (G,G′) in the symplectic group Sp(W) with G compact and let (G˜,G˜′) be the preimages of G and G′ in the metaplectic group Sp(W)˜. For every irreducible representation Π of G˜ appearing in Howe correspondence, we compute explicitly the restriction of the character ΘΠ′ of the associated representation Π′ of G˜′ on the set of regular points on the compact Cartan subgroup H˜′ of G˜′.

Symplectic group methods and the Arthurs Kelly model of measurement in quantum mechanics

We study the use of methods based on the real symplectic groups Sp(2n,R) in the analysis of the Arthurs-Kelly model of proposed simultaneous measurements of position and momentum in quantum mechanics. Consistent with the fact that such measurements are in fact not possible, with arbitrary precision, we show that the observable consequences of the Arthurs-Kelly interaction term are contained in the symplectic transformation law connecting the system plus apparatus variance matrices at an initial and a final time. The individual variance matrices are made up of averages and spreads or uncertainties for single hermitian observables one at a time, which are quantum mechanically well defined. The consequences of the multimode symplectic covariant Uncertainty Principle in the Arthurs-Kelly context are examined.

Classifying accidental symmetries in multi-Higgs doublet models

The potential of $n$-Higgs Doublet Models ($n$HDMs) contains a large number of SU(2)$_L$-preserving accidental symmetries as subgroups of the symplectic group Sp(2$n)$. To classify these, we introduce prime invariants and irreducible representations in bilinear field space that enable us to explicitly construct accidentally symmetric $n$HDM potentials. We showcase the classifications of symmetries and present the relationship among the theoretical parameters of the scalar potential for; (i) the Two Higgs Doublet Model (2HDM) and (ii) the Three Higgs Doublet Model (3HDM). We recover the maximum number of $13$ accidental symmetries for the 2HDM potential and for the first time, we present the complete list of $40$ accidental symmetries for the 3HDM potential.

Generation of the symplectic group by involutions

Let (V,b) be a symplectic space over a field where −1 is a square. Then, except for small dimensions, every element in the symplectic group Sp(V) is a product of 6 or fewer involutions.

The Segal-Bargmann transform on classical matrix Lie groups

We study the complex-time Segal-Bargmann transform Bs,τKN on a compact type Lie group KN, where KN is one of the following classical matrix Lie groups: the special orthogonal group SO(N,R), the special unitary group SU(N), or the compact symplectic group Sp(N). Our work complements and extends the results of Driver, Hall, and Kemp on the Segal-Bargman transform for the unitary group U(N). We provide an effective method of computing the action of the Segal-Bargmann transform on trace polynomials, which comprise a subspace of smooth functions on KN extending the polynomial functional calculus. Using these results, we show that as N→∞, the finite-dimensional transform Bs,τKN has a meaningful limit Gs,τ which can be identified as an operator on the space of complex Laurent polynomials.

Root systems and the Atiyah-Sutcliffe problem

In this short note, we show that the Atiyah-Sutcliffe conjectures for n = 2m, related to the unitary groups U(2m), imply the author’s analogous conjectures, which are associated with the symplectic groups Sp(m). The proof is based on the simple fact that the root system of U(2m) dominates that of Sp(m).

𝜔+-Type Index of GL+(2)-Paths

Abstract In this paper, we establish an ω + {\omega^{+}} -type index theory for paths in the general linear group GL + ⁢ ( 2 ) {\mathrm{GL}^{+}(2)} . This is done by the complete homotopy classification for such paths. We also compare this index theory with the ω index theory for paths in the symplectic group Sp ⁢ ( 2 ) {\mathrm{Sp}(2)} and obtain a generalization of Bott formula for iterated paths in GL + ⁢ ( 2 ) {\mathrm{GL}^{+}(2)} . As applications, the minimal periodic solution problem and the linear stability of general differential systems are studied.

Theta functions and a presentation of [formula omitted

We investigate a Curtis–Tits style group presentation based on the Dynkin diagram Cg, in which the short root generators have order two while the long root generators have order four. We prove that this describes a finite group with an almost extraspecial normal subgroup of order 22g+2 and quotient isomorphic to the symplectic group Sp(2g,2). Such an extension has to be non-split if g⩾3, and was proved to exist in papers of Bolt, Room and Wall from 1961/2 and Griess from 1973. Our presentation proves that it is a double cover of a finite quotient of Sp(2g,Z). We investigate a 2g dimensional complex representation on a suitable space of theta functions, and produce some consequences for the signatures of 4-manifolds described as surface bundles over surfaces. In particular, we prove that if the monodromy is contained in the theta subgroup Spq(2g,Z)⩽Sp(2g,Z) then the signature of the 4-manifold is divisible by eight.

String structures associated to indefinite Lie groups

String structures have played an important role in algebraic topology, via elliptic genera and elliptic cohomology, in differential geometry, via the study of higher geometric structures, and in physics, via partition functions. We extend the description of String structures from connected covers of the definite-signature orthogonal group O(n) to the indefinite-signature orthogonal group O(p,q), i.e. from the Riemannian to the pseudo-Riemannian setting. This requires that we work at the unstable level, which makes the discussion more subtle than the stable case. Similar, but much simpler, constructions hold for other noncompact Lie groups such as the unitary group U(p,q) and the symplectic group Sp(p,q). This extension provides a starting point for an abundance of constructions in (higher) geometry and applications in physics.

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.