Irreducible representation

In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation ( ρ , V ) {displaystyle ( ho ,V)} or irrep of an algebraic structure A {displaystyle A} is a nonzero representation that has no proper subrepresentation ( ρ | W , W ) , W ⊂ V {displaystyle ( ho |_{W},W),Wsubset V} closed under the action of { ρ ( a ) : a ∈ A } {displaystyle { ho (a):ain A}} . In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation ( ρ , V ) {displaystyle ( ho ,V)} or irrep of an algebraic structure A {displaystyle A} is a nonzero representation that has no proper subrepresentation ( ρ | W , W ) , W ⊂ V {displaystyle ( ho |_{W},W),Wsubset V} closed under the action of { ρ ( a ) : a ∈ A } {displaystyle { ho (a):ain A}} . Every finite-dimensional unitary representation on a Hermitian vector space V {displaystyle V} is the direct sum of irreducible representations. As irreducible representations are always indecomposable (i.e. cannot be decomposed further into a direct sum of representations), these terms are often confused; however, in general there are many reducible but indecomposable representations, such as the two-dimensional representation of the real numbers acting by upper triangular unipotent matrices. Group representation theory was generalized by Richard Brauer from the 1940s to give modular representation theory, in which the matrix operators act on a vector space over a field K {displaystyle K} of arbitrary characteristic, rather than a vector space over the field of real numbers or over the field of complex numbers. The structure analogous to an irreducible representation in the resulting theory is a simple module. Let ρ {displaystyle ho } be a representation i.e. a homomorphism ρ : G → G L ( V ) {displaystyle ho :G o GL(V)} of a group G {displaystyle G} where V {displaystyle V} is a vector space over a field F {displaystyle F} . If we pick a basis B {displaystyle B} for V {displaystyle V} , ρ {displaystyle ho } can be thought of as a function (a homomorphism) from a group into a set of invertible matrices and in this context is called a matrix representation. However, it simplifies things greatly if we think of the space V {displaystyle V} without a basis. A linear subspace W ⊂ V {displaystyle Wsubset V} is called G {displaystyle G} -invariant if ρ ( g ) w ∈ W {displaystyle ho (g)win W} for all g ∈ G {displaystyle gin G} and all w ∈ W {displaystyle win W} . The restriction of ρ {displaystyle ho } to a G {displaystyle G} -invariant subspace W ⊂ V {displaystyle Wsubset V} is known as a subrepresentation. A representation ρ : G → G L ( V ) {displaystyle ho :G o GL(V)} is said to be irreducible if it has only trivial subrepresentations (all representations can form a subrepresentation with the trivial G {displaystyle G} -invariant subspaces, e.g. the whole vector space V {displaystyle V} , and {0}). If there is a proper non-trivial invariant subspace, ρ {displaystyle ho } is said to be reducible. Group elements can be represented by matrices, although the term 'represented' has a specific and precise meaning in this context. A representation of a group is a mapping from the group elements to the general linear group of matrices. As notation, let a, b, c... denote elements of a group G with group product signified without any symbol, so ab is the group product of a and b and is also an element of G, and let representations be indicated by D. The representation of a is written By definition of group representations, the representation of a group product is translated into matrix multiplication of the representations: If e is the identity element of the group (so that ae = ea = a, etc.), then D(e) is an identity matrix, or identically a block matrix of identity matrices, since we must have and similarly for all other group elements. The last two staments correspond to the requirement that D is a group homomorphism.