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A decomposition into irreducible summands need not be unique even when the isotypic decomposition is
Remark
Theorem The isotypic decomposition of a completely reducible representation is unique identifies the canonical part of a semisimple decomposition: the block attached to each irreducible type .
What it does not canonically determine is a decomposition of one isotypic block into particular irreducible summands. Inside a block with multiplicity bigger than one, different complementary copies of the same irreducible can be chosen. The two-dimensional trivial representation of already shows this: both and split the same isotypic block into irreducible summands in different ways.
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Sources
- Peter Webb, A Course in Finite Group Representation Theory, Corollary 1.2.7 (standard reference, not scraped)