Abstract
Given a saturated fusion system over a finite p-group S, we provide criteria to determine when uniqueness of factorization into irreducible -invariant representations holds. We use them to prove uniqueness of factorization when the order of S is at most p 3. We also describe an example where the monoid of fusion-invariant representations is not even half-factorial. Finally, we find other examples of fusion systems where this monoid is not factorial using GAP.
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