COMBINATORIAL STRUCTURE OF STATE VECTORS IN UN .I. HOOK PATTERNS FOR MAXIMAL AND SEMIMAXIMAL STATES IN UN

It is shown that, in the boson-operator realization, the state vectors of the unitary groups Un - in the canonical chain Un ⊃ Un-1 ⊃ ⋯ ⊃ U1 - can be obtained ab initio by a combinatorial probabilistic method. From the Weyl branching law, a general state vector in Un is uniquely specified in the canonical chain; the algebraic determination of such a general state vector is in principle known (Cartan-Main theorem) from the state vector of highest weight; the explicit procedure is a generalization of the SU(2) lowering-operator technique. The present combinatorial method gives the normalization of these state vectors in terms of a new generalization of the combinatorial entity, the Nakayama hook, which generalization arises ab initio from a probabilistic argument in a natural way in the lowering procedure. It is the advantage of our general hook concept that it recasts those known algebraic results into a most economical algorithm which clarifies the structure of the boson-operator realization of the U n representations.