Symplectic Group Sp Research Articles

Right eigenvalues for quaternionic matrices: A topological approach

We apply the Lefschetz Fixed Point Theorem to show that every square matrix over the quaternions has right eigenvalues. We classify them and discuss some of their properties such as an analogue of Jordan canonical form and diagonalization of elements of the compact symplectic group Sp( n ).

Construction of p −1 Irreducible Modules with Fundamental Highest Weight for the Symplectic Group in Characteristic p

Let K be a field and let V be a vector space of dimension 2 m over K . Let ∧ V denote the exterior algebra of V and ∧ k V its k th exterior power for 0[les ] k [les ]2 m . Let f be a non-degenerate alternating bilinear form defined on V × V . The symplectic group Sp 2 m ( K ) is the group of all isometries of f and it acts as a group of vector space automorphisms on ∧ k V . In the case that K is algebraically closed and 1[les ] k [les ] m , it is known that ∧ k V contains a composition factor corresponding to the fundamental weight ω k of a root system of type C m . We shall refer to the irreducible module for Sp 2 m ( K ) given by this composition factor as a fundamental module.

Howe Correspondence for Real Unitary Groups

Roger Howe proved that for any reductive dual pair (G,G′) in the symplectic groupSp(2n,R), there is a one-to-one correspondence of irreducible admissible representations of some two-fold covers ofGandG′. We determine this correspondence explicitly for dual pairs of the form (U(p,q),U(r,s)) withr+s=p+q, and describe it in terms of Langlands parameters. In this case, the correspondence may be understood in a natural way as a correspondence of representations of the linear groups, rather than the appropriate covers. We show that every irreducible admissible representation ofU(p,q) occurs in the correspondence with precisely one unitary group of equal rank. This result verifies a conjecture of Harris, Kudla, and Sweet, who investigated the correspondence for unitary groups of equal size overp-adic fields. The correspondence of discrete series representations was determined by J.-S. Li. For induced representations, the correspondence is obtained in a natural way from the corresponding discrete series on the respective Levi factors of the parabolic subgroups ofU(p,q) andU(r,s). Generalizing a result of Li, we show that under the correspondence representations with nonzero cohomology are matched in an interesting way, with unitarity not necessarily preserved. The proof uses the induction principle which is due to Kudla, and an argument involvingK-types and the space of joint harmonics (Howe).

Branching rules for restriction of the Weil representations of Sp(n,R) to its maximal parabolic subgroup CM(n)

The symplectic group Sp(n,R) is the group of linear canonical transformations of a real 2n-dimensional phase space and CM(n)⊂Sp(n,R) is a maximal parabolic subgroup. The symplectic groups are the fundamental dynamical groups of classical and quantal Hamiltonian mechanics. In particular, Sp(3,R) is the dynamical group of the spherical harmonic oscillator and its Weil (harmonic series) representations are important for the microscopic (shell model) description of the collective motions of many-particle systems. The subgroup CM(3)⊂Sp(3,R) also appears in the microscopic theory of nuclear collective motion as the dynamical group of a hydrodynamic model of quadrupole vibrations and rotations of a nucleus. Thus, the Sp(3,R)→CM(3) branching rules are needed in finding the embedding of the hydrodynamic collective model in the microscopic shell model. Some new developments are made in the vector-coherent-state theory of induced representations.

Article

The role of Lie groups and algebras in symmetry-based models of the genetic code is considered. The two schemes, based upon the symplectic group Sp(6) and the exceptional group G(2) are shown to correspond to different embeddings in the group SO(14). Some possible alternative schemes are sketched. Problems with considering codons being represented as fermionic or bosonic are noted. A complete listing is given of all 64-dimensional irrecducible representations that can arise in the symmetric and alternating groups. PACS Nos.: 87.10 and 02.20

Once more nice equations for nice groups

In a previous paper, nice quintinomial equations were given for unramified coverings of the affine line in nonzero characteristic p p with the projective symplectic isometry group PSp ( 2 m , q ) (2m,q) and the (vectorial) symplectic isometry group Sp ( 2 m , q ) (2m,q) as Galois groups where m > 2 m>2 is any integer and q > 1 q>1 is any power of p p . Here we deform these equations to get nice quintinomial equations for unramified coverings of the once punctured affine line in characteristic p p with the projective symplectic similitude group PGSp ( 2 m , q ) (2m,q) and the (vectorial) symplectic similitude group GSp ( 2 m , q ) (2m,q) as Galois groups.

Hamiltonian Sp(3) BRST symmetry: existence theorems

The possibility of the extension of the BRST symmetry to the symplectic group sp(3) for the dynamical systems with first class independent constraints is proved in the Hamiltonian approach. The connection between the gauge fixing procedure into this larger symmetry and those used by the standard BRST theory is discussed.

Hermitian Metric Rigidity for Foliated Vector Bundles

We show that the canonical isometric imbedding of the symplectic group Sp(n) into R4n2 gives the least-dimensional isometric imbedding into the Euclidean space, even in the local standpoint. We prove this result by calculating the quantity pG determined by the curvature of Sp(n), which serves as an obstruction to the existence of local isometric imbeddings. We also exhibit the estimates on the value pG for the remaining compact classical simple Lie groups, and improve the previous results on the codimension of local isometric imbeddings of these groups.

Local Isometric Imbeddings of Symplectic Groups

We show that the canonical isometric imbedding of the symplectic group Sp(n) into R4n2 gives the least-dimensional isometric imbedding into the Euclidean space, even in the local standpoint. We prove this result by calculating the quantity pG determined by the curvature of Sp(n), which serves as an obstruction to the existence of local isometric imbeddings. We also exhibit the estimates on the value pG for the remaining compact classical simple Lie groups, and improve the previous results on the codimension of local isometric imbeddings of these groups.

Interpretation of Higher Order Magnetic effects in the Spectra of Transition Metal Ions in Terms of SO(5) and Sp(10)

Small discrepancies in the fitted energy levels of the configurations ${3d}^{N}$ of transition metal ions are ascribed to effective three-electron magnetic operators ${y}_{i}$. Surprisingly it has been found that, of the 16 possible operators with ranks 1 in both spin and orbital spaces, four operators labeled by the irreducible representation (irrep) (11) of SO(5) are sufficient to obtain results which appear to be limited by the errors in the experimental energy levels. An interpretation is given involving products of operators labeled by the irreps of SO(5) and the symplectic group Sp(10).

On the generators of SP n ( [formula omitted

We give a short variant of the classical proofs for the generation of the symplectic group Sp n ( Z ) over Z by the standard matrices.

Contraction of Exterior Powers in Characteristic 2 and the Spin Module

Let K be a field of characteristic 2 and letV be a vector space of dimension 2m over K. Let f be a non-degenerate alternating bilinear form defined on V × V. The symplectic group Sp(2m, K) acts on the exterior powers Λ k V for 0 ≤ k.≤ 2m There is a contraction map ∂ defined on the exterior algebra Λ, which commutes with the Sp(2m, K) action and satisfies ∂2 = 0 and ∂(Λ k V) ≤ Λ k−1 V We prove that ∂(Λ k V)= ker ∂ ∩ Λ k−1 V except when k=m+2. In the exceptional case, ∂(Λ m+2 V) has codimension 2m in ker ∂ ∩ Λ m V and we show that the quotient module ker ∂ ∩ Λ m V/∂ ∩ Λ m+2 V is a spin module for Sp(2m,K). When K is algebraically closed, we show that this spin module occurs with multiplicity 1 in Λ m V and multiplicity 0 in all other components of ΛV.

Plethysm for the non-compact group and newS-function identities

Methods of computing plethysms of the fundamental unitary irreducible representations of the non-compact symplectic group Sp(2n,R) are considered. Complete results are given for the symmetrized second powers. A number of new S-function identities are reported. The stability properties of the Sp(2n,R) plethysms are noted as well as a remarkable conjugacy relation. The application of the plethysms to N-particles in an isotropic harmonic oscillator is briefly outlined.

Theta dichotomy for unitary groups

Some recent work of Gross and Prasad [14] suggests that the root numbers attached to certain symplectic representations of the Weil-Deligne group of a local field F control certain branching rules for representations of orthogonal groups over F . On a global level, this local phenomenon should have implications for the structure of the value at the center of symmetry for certain L-functions of arithmetic interest [12]. This conjectural picture is based, to some extent, on the now classic work of Tunnell [47] and Waldspurger [49] as well as on the case of the triple product L-function [13, 34, 17]. In all of these examples, the local root number detects the existence of a certain type of invariant linear functional. It is possible to set up analogous conjectures for the isometry groups of Hermitian or quaternion-hermitian forms. It turns out that root numbers also play a role in the local and global theta correspondence. Roughly speaking, certain local root numbers should control the occurrence of representations in the local theta correspondence between groups of the ‘same size’, e.g., for dual pairs of the form (Sp(n), O(2n+1)) and (U(n), U(n)). As will be seen, the theta correspondence for such pairs is connected with a certain induced representation In(s, χ) at the point s = 0 on the unitary axis. By contrast, the correspondence for the pairs (Sp(n), O(2n)) and (Sp(n), O(2n+ 2)), discussed by Prasad [35], is connected to the behavior of a similar induced representation at the points ± 12 , in which case no epsilon factor arises. In this paper we consider the local theta correspondence for unitary groups in the non-archimedean case. Let F be a non-archimedean local field of characteristic not equal to 2, and let E be a quadratic extension of F . For further notation, see the notation section at the end of this introduction. Let V and W be E vector spaces of dimensions m and n equipped, respectively, with a Hermitian form ( , ) : V × V −→ E and a skew-Hermitian form 〈 , 〉 : W ×W −→ E. Then the isometry groups G(V ) and G(W ) of the spaces V and W form a dual reductive pair in the symplectic group Sp(W), where W = V ⊗E W , viewed as an F vector space of dimension 2mn and equipped with the symplectic form

Symplectic symmetry of Hubbard model and η pairing

By means of symplectic group Sp(2M), the symplectic symmetry of the Hubbard model and the η pairing are investigated to approach the eigenstates which have the largest spin multiplicities. The behavior of the off-diagonal long-range order of these eigenstates is examined by the criterion proposed by Yang and the energy diagram of the corresponding eigenvalues is elucidated by the η pairing mechanism. A basis set which is expressible in terms of η and ηa pairings is proposed.

Group-theoretic obstructions to integrability

AbstractLet V be a four-dimensional complex symplectic vector space. This paper classifies those connected linear algebraic subgroups of the symplectic group Sp(V) that admit two independent rational invariants. As an application we show the non integrability of a three degree of freedom Hamiltonian system.

Orthogonal Maslov index

0. Let us start with a few words about the ordinary Maslov index [1]. Given a~ R-vector space V with a symplectic form C(vl , v2) and a triple of Lagrangian subspaces L~ = Li C V, i = 1, 2, 3, we define an integer/~(L1, L2, L3) as follows. Let us consider another R-vector space L1 @ L2 @ L3 with the quadratic form B( l l , l~, ~) := C(~, ~) + C(~_, l~) + C(~3, I1). The Maslov index /~(L~, L2,L~) is defined as sign(B). The Maslov index implies an explicit formula for a discrete cocycle of the symplectic group Sp(V, C) as follows. Fix a Lagrangian subspace Lo C V. For g~,g2 E Sp(V, C) we put

GROUP THEORY CLASSIFICATION SCHEMES OF THE N-ELECTRON QUANTUM DOT

We develop a general framework for discussing collective behavior in confined many-electron systems. Our specific goal is the application to N-electron quantum dots, which are mesoscopic semiconductor systems of great current interest as possible ultra-small electronic devices. In view of its broad applicability, we are able to cast the discussion of the many-electron problem in general terms. We consider the general N-interacting particle system in d dimensions and study its bilinear dynamical symmetry group which is the noncompact symplectic group Sp(2Nd, R). Giving their explicit dependence on N and d, we focus on the classification of many-particle bound states which requires knowledge of the unitary discrete series representation theory of Sp(2Nd, R) and the corresponding character reductions. We also discuss matrix elements of the generators, the implementation of the Pauli principle, and a procedure to derive total angular momentum quantum numbers associated with a given total spin.

Products of transvections in one conjugacy class in the symplectic group over GF(3)

The transvections in any one conjugacy class Λ generate the symplectic group Sp(2 n, 3) over GF(3). The elements in Λ are called λ-transvections. There is a partition of the symplectic group into λ-standard, λ-exceptional, and λ-sporadic transformations. There are two conjugacy classes of λ-standard, three conjugacy classes of λ-exceptional, and seven classes of λ-sporadic transformations. We determine the λ-length of each symplectic transformation A, i.e., we determine l λ ( A) = t such that there are λ-transvections T i with A = T 1 ⋯ T t , but A is not a product of fewer than t λ-transvections.

Quantum-noise matrix for multimode systems: U(n) invariance, squeezing, and normal forms.

We present a complete analysis of variance matrices and quadrature squeezing for arbitrary states of quantum systems with any finite number of degrees of freedom. Basic to our analysis is the recognition of the crucial role played by the real symplectic group Sp(2n,openR) of linear canonical transformations on n pairs of canonical variables. We exploit the transformation properties of variance (noise) matrices under symplectic transformations to express the uncertainty-principle restrictions on a general variance matrix in several equivalent forms, each of which is manifestly symplectic invariant. These restrictions go beyond the classically adequate reality, symmetry, and positivity conditions. Towards developing a squeezing criterion for n-mode systems, we distinguish between photon-number-conserving passive linear optical systems and active ones. The former correspond to elements in the maximal compact U(n) subgroup of Sp(2n,openR), the latter to noncompact elements outside U(n). Based on this distinction, we motivate and state a U(n)-invariant squeezing criterion applicable to any state of an n-mode system, and explore alternative ways of expressing it. The set of all possible quantum-mechanical variance matrices is shown to contain several interesting subsets or subfamilies, whose definitions are related to the fact that a general variance matrix is not diagonalizable within U(n). Definitions, characterizations, and canonical forms for variance matrices in these subfamilies, as well as general ones, and their squeezing nature, are established. It is shown that all conceivable variance matrices can be generated through squeezed thermal states of the n-mode system and their symplectic transforms. Our formulas are developed in both the real and the complex forms for variance matrices, and ways to pass between them are given.