Symplectic Group Sp Research Articles

Low codimensional embeddings of Sp(n) and Su(n)

In [4] Elmer Rees proves that the symplectic group Sp(n) can be smoothly embedded in Euclidean space with codimension 3n, and the unitary group U(n) with codimension n. These are special cases of a result he obtains for a compact connected Lie group G. The general technique is first to embed G/T, where T is a maximal torus, as a maximal orbit of the adjoint representation of G, and then to extendto an embedding of G by using a maximal orbit of a faithful representation of G. In thisnote, we observe that in the cases G = Sp(n) or SU(n) an improved result is obtained byusing the “symplectic torus” S3 x … x S3 in place of T = S1 x … x S1. As in Rees's construction, the normal bundle of the embedding of G is trivial.

Abelian Unipotent Subgroups of Finite Unitary and Symplectic Groups

AbstractIf G is the unitary group U(V) or the symplectic group Sp(V) of a vector space V over a finite field of characteristic p, and r is a positive integer, we determine the abelian p-subgroups of largest order in G whose fixed subspaces in V have dimension at least r, with the restriction that we assume p ≠ 2 in the symplectic case. In particular, we determine the abelian subgroups of largest order in a Sylow p-subgroup of G. Our results complement earlier work on general linear and orthogonal groups.

Interacting two-vector-boson model of collective motions in nuclei

A boson representation of the Hamiltonian is introduced to describe collective motions in nuclei. Two interacting vector bosons differing in their 'pseudospin' projections are used. The non-compact symplectic group Sp(12, R) is the group of dynamical symmetry for the Hamiltonian. If the Hamiltonian satisfies the restriction of preservation of the number of bosons, then in the basis of Sp(12, R) its matrix is block-diagonal with respect to the maximal compact subgroup of Sp(12, R), namely the group U(6). The U(6) structure of the number-preserving Hamiltonian is investigated. The two-boson interaction is expressed in terms of the second-order Casimir operators of the various chains of subgroups of U(6).

On the theory of collective motion in nuclei. II. Quantum theory

The collective Hamiltonian is assumed to be invariant under the orthogonal group O(A − 1, R ). It is shown that the quantum collective dynamics can be formulated on a coset space of the symplectic group sp (6, R ) of dimension 12, 16 or 18. The first case corresponds to the collective dynamics based on closed shells and leads to a Hilbert space of analytic functions in six complex collective quasiparticle variables. Dequantization of this scheme yields the classical dynamics described in I. In the limit A ⪢ 1 one obtains a system of s- and d-bosons with symmetry group u (6) in the collective state space.

Generators of Sp n( V) over a quasi semilocal semihereditary ring

This paper is devoted to determine the minimal length of expressions of an isometry in a symplectic group Sp n ( V) by a product of transvections under the assumption that V is an n-ary nonsingular alternating space over a quasi semilocal semihereditary ring with 2 as a unit.

Basis states for equivalent electrons. III. States for the Lie group SO(2l + 1) × SU(2) using Sp(4l + 2) states

The Lie group SO(2l + 1) × EU(2) is a subgroup of the symplectic group Sp(4l + 2), which in turn is a subgroup of the unitary group U(4l + 2). The Slater basis states for N equivalent electrons form the basis for the irreducible representation (1N) of U(4l + 2). The basis states for the irreducible representations of SO(2l + 1) × SU(2) are expressed in terms of the states for irreducible representations of Sp(4l + 2). The basis states for SO(2l + 1) × SU(2) are also expressed in terms of the Slater basis states.

The cranked oscillator coherent states

The oscillator cranking-model wavefunction can be identified with a set of coherent states of the symplectic group Sp(4,R). A quantum-classical correspondence for the model is, however, established through a set of coherent states of the Heisenberg symplectic group N(2)(X)Sp(4,R). The properties of these states and their usefulness in various fields of physics are studied in detail. Interesting and useful generalisations of these coherent states are also discussed.

The most degenerate irreducible representations of the symplectic group

Matrix elements of the generators of the symplectic group Sp(2n) have been obtained for the most degenerate irreducible representations (m2n,?) and (?2n). These are the only two cases for n≳2 where the state labels according to the branching laws of Hegerfeldt give rise to an orthogonal set of basic vectors.

Products of reflections in the quaternionic unitary group

Let Sp( n) denote the unitary group of a positive definite hermitian form on an n-dimensional right quaternionic vector space. Thus Sp( n) is the compact real form of the complex symplectic group Sp(2 n, C). For each A ϵ Sp( n), n ⩾ 2, we determine the smallest number m such that A can be written as a product of m reflections in Sp( n). In particular, we show that if n ⩾ 4 then every A ϵ Sp( n) is a product of n, n + 1, or n + 2 reflections. In the projective group PSp( n), n > 4, every element is a product of n reflections.

Nilpotency in classical groups over a field of characteristic 2

Let V be a form space. If W is a subspace of V we put W {ve Vl~(v, W) =0}. The form space V is called non-defective if V==0. It is called nondegenerate if dim(V=)< 1 and ~(v)#:0 for all non-zero wV  1.2. Assume that V is non-degenerate. V is defective if and only if char(k)=2 and e=0 and dim(V) is odd. If e= 1 then dim(V) is even. Let G = G(V) be the algebraic group in the sense of [-1] of the automorphisms geGl(V) which leave the forms fi and c~ invariant. Let g = g(V) be its Lie algebra. If e = 1 then G is the symplectic group Sp(V), so it is connected and semi-simple. If e=0 then G is the orthogonal group O(V), which is reductive. O(V) has two components if V is non-defective. O(V) is connected if V is defective. 1.3. Our object is to describe the conjugacy classes of the unipotent elements of G and of the nilpotent elements of g under the adjoint action of G given by

On the accidental degeneracy of the n-dimensional anisotropic harmonic oscillator. II

In an earlier paper, a group, G, associated with the n-dimensional anisotropic harmonic oscillator was shown to be embedded in a semidirect product, L, of the Weyl group N and the symplectic group Sp(2n,R). A particular representation Rv of L, when restricted to G, was proved to be unitarily equivalent to ⊕sdω,s UGv,−(sgnv)s), where dω,s is the degeneracy of the energy level Eω,s of the n-dimensional anisotropic harmonic oscillator with frequencies (ω1,ω2,...,ωn) =ω, UGv,−(sgnv)s is an irreducible representation of G and s may be regarded as indexing all distinct energy levels of the system. In the present paper, the representation Rv of L is shown to be unitarily equivalent to the representation UNvW⊕? of L, where UNv is an irreducible representation of N, W is the projective representation of Sp(2n,R) which intertwines the representations UNv and SUNv of N [where S ε Sp(2n,R)], and ? is the complex conjugate of W. This alternative form for the representation Rv of L enables it to be decomposed, into two irreducible representations.

On the accidental degeneracy of the anisotropic harmonic oscillator. I

A group G associated with the n-dimensional anisotropic harmonic oscillator is shown to be embedded in a semidirect product L of the Weyl group N and the symplectic group Sp(2n,R). A particular induced representation of the group L, when restricted to G, is proved to be unitarily equivalent to ⊕sdω,sUGv,−(sgnv)s) where dω,s is the degeneracy of the energy level Eω,s of the n-dimensional anisotropic harmonic oscillator with frequencies (ω1, ω2,...,ωn) =ω, UGv,−(sgnv)s is an irreducible representation of G, and s may be regarded as indexing all distinct energy levels of the system.

Canonical realizations of the lie algebrasp(2n, R)

The generators of the Lie algebra of the symplectic groupsp(2n, R) are, recurrently, realized by means of polynomials in the quantum canonical variablesp i andq i. These realizations are skew-Hermitian, the Casimir operators are realized by constant multiples of identity elements, and, depending on the number of the canonical pairs used, they depend ond, d=1, 2, ...,n free real parameters.

The generation of Sp( [formula omitted]2) by transvections

In 1955, in [2], Dieudonne, observing that each symplectic group Sp(V) is generated by its transvections, considered the problem of finding the minimal number of factors in the expression of elements 0 of Sp(V) as products of transvections in Sp(V).l The answer is in general residue cr, but with some classes of exceptions. Complications arise when the underlying field is 5, , and Dieudonne’s list of exceptions [2, p. 1641 for this case is incomplete. The difficulty occurs in 4.4” page 163 and seems to lie in the fact that, following the notation there, the plane T chosen orthogonal to c and d need not (in fact, cannot) be orthogonal to the revised “c” and ‘Id,” as is tacitly assumed. The proof in 4.4” fails and there is a class of exceptions of residue 5 in dimension 6 (see 3.3 below). This paper is devoted to completing the list of exceptions for ffs . Also mentioned are the minor corrections

The principal series ofSp(n, ℝ)

The principal series of unitary representations of the noncompact symplectic groupSp(n, ℝ) is constructed for alln. The Lie algebra ofSp(n, ℝ) is isomorphic to the algebra of bilinear products of boson operators inn dimensions. The spectrum of the number operator for the principal series representations is shown to be unbounded, both from above and from below.

The symplecton: A prototype for semi-simple graded Lie algebras

An elementary, physically motivated, example of a semi-simple graded Lie algebra (SSGLA) is shown to be given by the ‘symplecton realization’ of angular momentum [an associative, involutive, inner-product algebra whose characteristic polynomials realize the symplectic group Sp(2)]. The Pais-Rittenberg result shows that the n-component symplecton realizes the most general, SSGLA, Sp(2n).

Automorphisms of the symplectic group over a local ring

Let R denote a commutative local ring with maximal ideal m and residue field K = R m . Let V be a symplectic space over R. In this paper we determine the group automorphisms of the symplectic group Sp n ( V) when n ⩾ 6, the characteristic of k is not 2, and k is not the finite field of three elements.

Lowering operators and the symplectic group

A set of operators is constructed which can be used for defining a basis in a finite dimensional irreducible representation space of the symplectic group Sp(2n). Each basis vector is an eigenvector of all invariant operators of the canonical subgroups U(1)⊂ ⊂Sp(2)⊂…⊂Sp(2n−2)⊂Sp(2n).

Linear Canonical Transformations and Their Unitary Representations

We show that the group of linear canonical transformations in a 2N-dimensional phase space is the real symplectic group Sp(2N), and discuss its unitary representation in quantum mechanics when the N coordinates are diagonal. We show that this Sp(2N) group is the well-known dynamical group of the N-dimensional harmonic oscillator. Finally, we study the case of n particles in a q-dimensional oscillator potential, for which N = nq, and discuss the chain of groups Sp(2nq)⊃Sp(2n)×O(q). An application to the calculation of matrix elements is given in a following paper.

Degenerate Representations of the Symplectic Groups II. The Noncompact Group Sp(p, q)

Using a method based on geometrical properties of homogeneous spaces of rank one homeomorphic to coset spaces of Lie groups, a series of degenerate unitary irreducible representations of the noncompact symplectic group Sp(p, q) is investigated. The representation spaces for a discrete series determined by two integer numbers and a continuous series determined by one real and one integer parameter are given, the corresponding basis functions being formed by the linear combinations of eigenfunctions of the Laplace-Beltrami operator of the considered space. Explicit formulas for the action of generators of Sp(p, q) in these representations are obtained. The results provide a deeper insight into the structure of the two-parameter ``not most degenerate'' unitary representations.