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A completely reducible representation as a finite direct sum of irreducible subrepresentations
Definition
Let be a finite-dimensional representation of over a field . The representation is completely reducible if there are irreducible subrepresentations such that
The empty direct sum is allowed, so the zero representation is completely reducible.
Under the dictionary of Under the dictionary, subrepresentations are exactly submodules and irreducible representations are exactly simple modules, this is exactly the representation-side form of a semisimple left -module (Semisimple modules as direct sums of simple modules).
Depends on
- A finite-dimensional representation $\rho:G\to \operatorname{GL}(V)$ over a field, and its degree
- Subrepresentations, direct sums of representations, and irreducibility
- Under the dictionary, subrepresentations are exactly submodules and irreducible representations are exactly simple modules
- Semisimple modules as direct sums of simple modules
Used by
- If char k ∤ |G|, every finite-dimensional representation of G is completely reducible Corollary
- If char k ∤ |G|, then k[G] is a semisimple ring Corollary
- The isotypic component of a completely reducible representation Definition
- The isotypic decomposition of a completely reducible representation is unique Theorem
Dependency tree · two levels
14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Peter Webb, A Course in Finite Group Representation Theory, Chapter 1 Section 1.2 (standard reference, not scraped)
- Pavel Etingof et al., Introduction to Representation Theory, Chapter 2 Section 2.1 (standard reference, not scraped)