Subrepresentations, direct sums of representations, and irreducibility

Depends on

• A finite-dimensional representation $\rho:G\to \operatorname{GL}(V)$ over a field, and its degree
• Linear subspace of a vector space

Used by

• If char k ∤ |G|, every finite-dimensional representation of G is completely reducible Corollary
• M-groups are solvable (Taketa) Corollary
• Under the dictionary, subrepresentations are exactly submodules and irreducible representations are exactly simple modules Corollary
• A reducible Specht module in characteristic two Counterexample
• A solvable group that is not an M-group Counterexample
• The branching filtration need not split in modular characteristic Counterexample
• A completely reducible representation as a finite direct sum of irreducible subrepresentations Definition
• An irreducible complex character Definition
• Supersolvable groups and monomial characters Definition
• The endomorphism division algebra of an irreducible representation Definition
• The isotypic component of a completely reducible representation Definition
• All three Specht modules of S₃ Example
• Convolution of matrix coefficients on a compact group Example
• The real 2-dimensional irreducible representation of C₃ has endomorphism ring ℂ Example
• The standard 2-dimensional representation of S₃ inside the permutation representation on ℂ³ is irreducible Example
• The two-dimensional trivial representation of C₂ has many irreducible splittings but one isotypic component Example
• Trivial and one-dimensional monomial cases Example
• False statement: Maschke's theorem holds over every field False statement
• A coset basis makes a monomial representation monomial matrices Lemma
• Additive characters are exactly one-dimensional complex representation characters Lemma
• A faithful irreducible is induced from a proper inertia subgroup Proposition
• Complete reducibility of finite-dimensional compact-group representations Theorem
• Complex Specht modules are irreducible Theorem
• Every irreducible representation of a finite abelian group over a splitting field is one-dimensional Theorem
• Homomorphisms from Specht to Young permutation modules obey dominance Theorem
• James's submodule theorem over the complex numbers Theorem
• Maschke's theorem for finite groups over fields whose characteristic does not divide |G| Theorem
• Specht modules classify the complex irreducibles of Sₙ Theorem
• Specht restriction has a removable-corner filtration over every field Theorem

Dependency tree · two levels

11 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)