A finite-dimensional representation $\rho :G\to \mathrm{GL}(V)$ over a field, and its degree

Depends on

• Finite-dimensional vector space, and its dimension $\dim_F V$; infinite-dimensional means having no finite basis
• Field
• Monoid homomorphism and group homomorphism
• Invertible linear maps, linear isomorphisms, and inverse linear maps
• An $R$-linear action of $G$ on a left $R$-module, and a $G$-module over $R$
• Vector space over a field

Used by

• Every irreducible representation of a finite group has degree at most |G| Corollary
• The dimension of an induced finite-dimensional representation is [G:H]dim W Corollary
• A reducible Specht module in characteristic two Counterexample
• A solvable group that is not an M-group Counterexample
• A completely reducible representation as a finite direct sum of irreducible subrepresentations Definition
• A splitting field for a finite group: every irreducible representation has scalar endomorphism ring Definition
• Additive characters of a finite abelian group Definition
• An extension of a normal subgroup representation Definition
• Character fields and fields of definition Definition
• Commuting symmetric-group and linear actions on a tensor power Definition
• Conjugate representations and conjugate characters on conjugate subgroups Definition
• Galois conjugates of a representation Definition
• Intertwiners, the spaces Hom_G(V,W) and End_G(V), equivalent representations, and faithful representations Definition
• Monomial representations, monomial characters, and M-groups Definition
• Subrepresentations, direct sums of representations, and irreducibility Definition
• The character χ_V(g)=tr(ρ_V(g)) of a finite-dimensional complex representation Definition
• The dual or contragredient complex representation Definition
• The fixed subspace V^G of a representation Definition
• The rational representation ring R_ℚ(G) and rational-valued class functions Definition
• The restriction coproduct on the graded symmetric-group character ring Definition
• The sign representation of Sₙ and the restriction Res^G_H(V) of a representation to a subgroup Definition
• The skew multiplicity module K^λ/μ over ℂ Definition
• The tensor product of two complex representations Definition
• The trivial representation, the regular representation, and permutation representations from finite G-sets Definition
• A circle representation with an averaged orthogonal weight form Example
• A compact group with no faithful continuous finite-dimensional representation Example
• All three Specht modules of S₃ Example
• Convolution of matrix coefficients on a compact group Example
• The quaternion group Q₈ acts on ℍ by left multiplication Example
• Trivial and one-dimensional monomial cases Example
• False statement: Maschke's theorem holds over every field False statement
• Additive characters are exactly one-dimensional complex representation characters Lemma
• Averaging a Hermitian form unitarizes a finite-dimensional compact-group representation Lemma
• Base change for intertwiner spaces Lemma
• Induction commutes with an external tensor factor Lemma
• Induction is invariant under conjugation of the subgroup and the representation Lemma
• Modular trace depends only on the p-regular part Lemma
• The cocycle central extension linearizes a projective representation Lemma
• The kernel of a finite direct sum is the intersection of the kernels Lemma
• Adjoint is a smooth Lie-group representation Proposition

…and 4 more results.

Dependency tree · two levels

29 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

• Pavel Etingof et al., Introduction to Representation Theory, Section 1.3 (standard reference, not scraped)