Alphabeta MathHow statement and proof provenance work
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The isotypic component of a completely reducible representation
Definition
Let be a completely reducible representation of a group over a field , and let be an irreducible representation of over . The isotypic component of of type is
If no irreducible subrepresentation of is equivalent to , this sum is .
Only the equivalence class of matters: if , then the defining collections of irreducible subrepresentations are the same, so .
Depends on
- A completely reducible representation as a finite direct sum of irreducible subrepresentations
- Subrepresentations, direct sums of representations, and irreducibility
- Intertwiners, the spaces $\operatorname{Hom}_G(V,W)$ and $\operatorname{End}_G(V)$, equivalent representations, and faithful representations
Used by
Dependency tree · two levels
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Sources
- Peter Webb, A Course in Finite Group Representation Theory, Corollary 1.2.7 (standard reference, not scraped)
- Pavel Etingof et al., Introduction to Representation Theory, Proposition 2.2 (standard reference, not scraped)