Irreducible Representations Of Sp Research Articles

An Elementary Approach to Weight Multiplicities in Bivariate Irreducible Representations of Sp(2r)

By bivariate irreducible representations of Sp(2r), we mean irreducible representations with highest weights containing at most two nonzero entries, using the usual identification of dominant weights for complex symplectic Lie algebras and their corresponding Lie groups as r-tuples in decreasing non-negative integers. This paper has two aims. The first aim is to provide a formula for the weight multiplicities of said representations, which is easily computable. The second aim is to present these weight multiplicities using elementary means. The formula for these weight multiplicities is derived using basic multiliear algebra and combinatorial arguments through explicit descriptions of weight vectors.

Degenerate principal series representations of Sp(p, q)

Letp>q and letG=Sp(p, q). LetP=LN be the maximal parabolic subgroup ofG with Levi subgroupL≅GL q (ℍ)×Sp(p−q). Forseℂ andμ a highest weight of Sp(p−q), let пs,µ be the representation ofP such that its restriction toN is trivial and $$\pi s,\mu |L = \det _q^s $$ ⊠T p-q μ , where det q is the determinant character of GL q (ℍ) andT p-q μ is the irreducible representation of Sp(p−q) with highest weightμ. LetI p,q(s, μ) be the Harish-Chandra module of the induced representation Ind G $$\begin{array}{*{20}c} G \\ P \\\end{array}\pi _{s,\mu } $$ . In this paper, we shall determine the module structure and unitarity ofI p, q(s, μ).

Representation theory of the symplectic groups. I

Certain aspects of symplectic group representation theory are investigated. In particular, it is shown that every irreducible representation of Sp(n) admits a relatively large class of states, referred to herein as canonical states, which possess properties analogous to the Gelfand–Tsetlin states appearing in the theory of the orthogonal and unitary groups. The properties of canonical states are investigated and some matrix element formulas are derived.

Polynomial representations of the symplectic groups

The irreducible finite dimensional representations of the symplectic groups are realized as polynomials on the irreducible representation spaces of the corresponding general linear groups. It is shown that the number of times an irreducible representation of a maximal symplectic subgroup occurs in a given representation of a symplectic group, is related to the betweenness conditions of representations of the corresponding general linear groups. Using this relation, it is shown how to construct polynomial bases for the irreducible representation spaces of the symplectic groups in which the basis labels come from the representations of the symplectic subgroup chain, and the multiplicity labels come from representations of the odd dimensional general linear groups, as well as from ℂ subgroups. The irreducible representations of Sp(4) are worked out completely, and several examples from Sp(6) are given.

Infinite-dimensional representations of the graded Lie algebra (Sp(4):4). Representation of the para-Bose operators with real order of quantization

Infinite-dimensional representations for the system of the two para-Bose operators, which generate the graded Lie algebra (GLA) (Sp(4):4), are constructed by using the irreducible representations (discrete series) of the Lie algebra Sp(4)≊SO(3,2). It is shown that there are four kinds of the irreducible representations of the GLA’s (Sp(4):4), i.e., cases (I), (II), (III), and (IV). Case (I) is described by the three irreducible representations of Sp(4) and corresponds to the para-Bose quantization with real order of the quantization greater than 1 and case (IV) corresponding to the ordinary Bose quantization by the two irreducible representations of Sp(4). Cases (II) and (III), which are described by the four and two irreducible representations of Sp(4), respectively, and cannot be obtained by the method of Fock space, express the representations of the graded Lie algebra (Sp(4):4).

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.

Basis states for equivalent electrons. II. States for the Lie group Sp(4l + 2)

The Slater basis states for N equivalent electrons form the basis for the irreducible representation (1N) of the Lie group U(4l + 2). States which are eigenfunctions of the total spin and total orbital angular momentum form the basis for irreducible representations of SO(3) × SU(2). In this paper the intermediate group Sp(4l + 2) is studied. The basis states for irreducible representations of Sp(4l + 2) are expressed in terms of the Slater basis states.

Symplectic Stiefel harmonics and holomorphic representations of symplectic groups

Let I k {I_k} denote the identity matrix of order k and set \[ s k = [ 0 a m p ; − I k I k a m p ; 0 ] . {s_k} = \left [ {\begin {array}{*{20}{c}} 0 & { - {I_k}} \\ {{I_k}} & 0 \\ \end {array} } \right ]. \] Let Sp ( k , C ) {\text {Sp}}(k,{\mathbf {C}}) denote the group of all complex 2 k × k 2k \times k matrices which satisfy the equation g s k g t = s k g{s_k}{g^t} = {s_k} . Let E be the linear space of all n × 2 k n \times 2k complex matrices with k ⩾ n k \geqslant n , and let S ( E ∗ ) S({E^\ast }) denote the symmetric algebra of all complex-valued polynomial functions on E. The study of the action of Sp ( k , C ) {\text {Sp}}(k,{\mathbf {C}}) , which is obtained by right translation on S ( E ∗ ) S({E^\ast }) , leads to a concrete and simple realization of all irreducible holomorphic representations of Sp ( k , C ) {\text {Sp}}(k,{\mathbf {C}}) . In connection with this realization, a theory of symplectic Stiefel harmonics is also established. This notion may be thought of as a generalization of the spherical harmonics for the symplectic Stiefel manifold.

The discrete series ofSp(n,ℝ)

For each irreducible representation [λ] ofU(n), a discrete irreducible unitary representation ofSp(n) is constructed for which the oscillator Hamiltonian is bounded from below with its ground-state eigenspace transforming according to [λ] underU(n). A basis of eigenstates for the harmonic oscillator is determined and the action of the Lie algebrasp(n) on that basis explicitly given. Connections with the Bohr collective vibrational model are established.

Construction of irreducible representations ofSp 4,R by means of para-boson operators

We give the complete solution to the problem of constructing IR's of the creation and destruction operators for a system of two para-oscillators for any order of para-statistics. By means of this result, we construct explicitly a class of unitary IR's ofSp4,R algebra, belonging to the discrete series.