Operators that Lower or Raise the Irreducible Vector Spaces of <i>U</i> <i>n</i>−1 Contained in an Irreducible Vector Space of <i>Un</i>

Abstract

We define operators that lower or raise the irreducible vector spaces of a semisimple subgroup of a semisimple Lie group contained in an irreducible vector space of the group. We determine the lowering and raising operators for the canonical subgroup Un−1 of the unitary group Un. With the help of these operators, which are polynomial functions of the generators of Un, and the corresponding operators for the subgroups in the canonical chain Un⊃ Un−1⊃ …⊃ U2⊃ U1 we can obtain, in this chain, the full set of normalized basis vectors of an irreducible vector space of Un from any given normalized basis vector of the vector space. In particular we can obtain, using only the lowering operators, the set of basis vectors from the basis vector of highest weight of the vector space. This result is of importance in applications to many-body problems and in the determination of the Wigner coefficients of Un. In future papers we plan to determine the lowering and raising operators for the orthogonal and symplectic groups.