Uniqueness and Multiplicity One of the Highest Weight identifies how weights organize finite-dimensional representations of compact Lie groups. In concrete terms, it formalizes the statement that let be a compact connected Lie group with gT G t t C^ g C t C^+ t C^V t C^V\0\ t C^ 00 t C^VV^+X( g C)(T,^+)$, and CV=1..