Irreducible representation

An irreducible representation is a nonzero representation with no proper nonzero invariant subspaces. It cannot be decomposed into smaller invariant representations.

Group representation: A group representation is a homomorphism from a group to the invertible linear transformations of a vector space. Irreducibility is a property of group representations and their invariant subspaces.
Maschke's theorem: Maschke's theorem states that finite-dimensional representations of finite groups are completely reducible when the field's characteristic does not divide the group order. It ensures such representations decompose into irreducible summands.
Reducible representation: A reducible representation has a proper nonzero invariant subspace. It is the direct opposite of irreducibility under the invariant-subspace definition.
Fourier analysis on finite groups: Fourier analysis on finite groups expands functions using matrix coefficients of irreducible representations. Irreducible representations provide the frequency components of functions on a finite group.
Ferdinand Georg Frobenius: Ferdinand Georg Frobenius was a German mathematician whose work helped establish the representation theory of finite groups. His character theory made irreducible representations of finite groups calculable.
Invariant subspace: An invariant subspace is a subspace preserved by every operator in a specified family of linear transformations. A representation is irreducible precisely when its only invariant subspaces are zero and the whole space.
Complete reducibility: Complete reducibility is the property that a representation decomposes as a direct sum of irreducible subrepresentations. It describes how irreducible representations assemble into larger representations.
Indecomposable representation: An indecomposable representation cannot be expressed as a direct sum of two nonzero subrepresentations. It may have proper invariant subspaces, unlike an irreducible representation.
Peter–Weyl theorem: The Peter–Weyl theorem decomposes square-integrable functions on a compact group into matrix coefficients of finite-dimensional irreducible representations. It extends the role of irreducibles from finite groups to compact groups.
William Burnside: William Burnside was a British mathematician who made foundational contributions to the theory of finite groups and their representations. His work helped shape the early systematic study of finite-group representations.

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