Combinatorial Structure of State Vectors in Un. I. Hook Patterns for Maximal and Semimaximal States in Un

Abstract

It is shown that, in the boson‐operator realization, the state vectors of the unitary groups U_n—in the canonical chain $U_n{\mathrm{\supset U}}_{\text{n−1}}{\mathrm{\supset \cdots \supset U}}_1$ —can be obtained ab initio by a combinatorial probabilistic method. From the Weyl branching law, a general state vector in U_n 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.