We analyze the criterion of the multiplicity-free theorem of representations [5, 6] and explain its generalization. The criterion is given by means of geometric conditions on an equivariant holomorphic vector bundle, namely, the ''visibility'' of the action on a base space (i.e. generic orbits intersecting with a real form) and the multiplicity-free property on a fiber.
Then, several finite dimensional examples are presented to illustrate the general multiplicity-free theorem, in particular, explaining that three multiplicity-free results stem readily from a single geometry in our framework. Furthermore, we prove that an elementary geometric result on Grassmann varieties and a small number of multiplicityfree results give rise to all the cases of multiplicity-free tensor product representations of GL(n, C), for which Stembridge  has recently classified by completely different and combinatorial methods.
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