Abstract
This paper concerns the enumeration of simultaneous conjugacy classes of tuples of commuting unitary matrices and of commuting symplectic matrices over a finite field of odd size. For any given conjugacy class, the orbits for the action of its centralizer group on itself by conjugation (that is, the conjugacy classes within the centralizer group) are called branches. We determine the branching rules for the unitary groups , and for the symplectic groups
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