Irreducible Representations of the Five-Dimensional Rotation Group. I

Abstract

Explicit ~atrix. eleme~~s are foun~ !~~ the ,generators of the ~roup R(5) in an arbitrary irreducible representatIOn usmg the natural basIs m whIch th.e represen.tatIon of R(5) is fully reduced with respect to the.subg~oup R(4) = .SU(~) @ SU(2). ,!,he t~chnIque used IS based on the well-known Racah algebra. The dlmen.slOn formula ~s derIved and the mvarIants are found. A family of identities is established which relates varIOUS polynomIals of degree four in the generators and which holds in any representation of the group.

A computationally tractable version of the collective model

“…A suitable minimal set of wave functions is found by examination of the angular momentum content of the L 2 (S 4 ) = v H (SO (5)) v space. Knowing the SO (5)→SO (3) branching rules for the irreps appearing in the collective model [28], it is possible to arrange the angular-momentum states of L 2 (S 4 ) into K bands, each having the same sequence of angular-momentum states as those of axially-symmetric rotor bands of the same K, and with the sequence of K bands being in one-to-one correspondence with those of a sequence of gamma-vibrational bands. More precisely, the band-heads of the K bands appear with increasing seniority v in the sequence 1 2 3 3 5 6 7 8 9 …”

GENERALIZATION OF THE GELL-MANN FORMULA FOR sl(5, ℝ) AND su(5) ALGEBRAS

“…The admissible values of J 1 and J 2 , within an irreducible representation (J 1 , J 2 ) are given in [22]. Now, the basis of the so(5) algebra, i.e.…”

Construction of spherical harmonics and Clebsch–Gordan coefficients

“…The "missing label" is provided by a multiplicity index α. The branching rule for SO(3) irreps occurring within an SO(5) irrep is well known [16,17], and the multiplicity is given by (A1). The SO(5) ⊃ SO(3) spherical harmonics have physical significance as the angular, i.e., (γ, Ω), wave functions for the nuclear collective model, for the case in which which the collective potential is SO(5) invariant [18,19].…”

“…obtained by direct application of (16). It follows that [33] If the SO(5) representations are labeled according to the SO(5) ⊃ SO(3) ⊃ SO(2) subalgebra chain (1), each coupling coefficient may be written as the product of an SO(3)-reduced Clebsch-Gordan coefficient and an ordinary SO(3) Clebsch-Gordan coefficient, according to the Racah factorization lemma [34] (see Ref.…”