What is a Group Representation and How Does it Act on a Vector Space?

[Greg Bernhardt]

Definition/Summary

A group representation is a realization of a group in the form of a set of matrices over some algebraic field, usually the complex numbers.

A representation is irreducible if the only sort of matrix that commutes with all its matrices is a sort that is proportional to the identity matrix. An irreducible representation is sometimes called an irrep. The number of irreps of a group is equal to its number of conjugacy classes.

One may decompose a reducible representation into irreps by transforming its matrices into block-diagonal matrices where each block is an irrep matrix:

D(a) -> {D1(a), D2(a), ..., Dn(a)}

Equations

Representation matrices D(a) for elements a satisfy

D(a).D(b) = D(a*b)

The identity one is D(e) = I, and the inverse is
[itex]D(a^{-1}) = D^{-1}(a)[/itex]

A representation can be transformed into an equivalent one with
[itex]D(a) \to SD(a)S^{-1}[/itex]
for some matrix S.

However, the traces of the representation matrices, the representation characters, remain unchanged.

Extended explanation

There are some interesting special cases of representations.

Every finite or countably-infinite group has a regular representation. This representation can be constructed as follows. Define an index function for each element i(a) for constructing the indices to the representation matrices. Those matrices are thus

D_i(x),i(ax)(a) = 1 for all x in the group, 0 otherwise

An irrep k with dimension n_k has n_k copies of it in the regular representation, and that gives this interesting expression for the group's order:
[itex]n_G = \sum_k (n_k)^2[/itex]

It can also be shown that n_k evenly divides the group's order.

The irreps of abelian groups have dimension 1, and those of finite abelian groups are the products of irreps of their component cyclic groups.

The irreps of cyclic group Z(n) are given as follows for element j and irrep k:
D^(k)(a^j) = ω^jk

where ω is an nth root of unity, a is a generator, and j and k range from 0 to n-1.

An irrep is self-conjugate if there is some matrix S that satisfies
D^*(a) = SD(a)S^-1
for all a in the group. If an irrep is not self-conjugate, it is complex.

If an irrep has det(S) = +1, then it is real, while if it has det(S) = -1, then it is pseudoreal or quaternionic. Every pseudoreal representation has even dimension.

* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!

 

[fresh_42]

See also https://www.physicsforums.com/insights/representations-precision-important/
Important to know is, that the wordings: representation - operates on - acts on are basically the same.

A linear representation of a group ##G## on a vector space ##V## is a group homomorphism ##G \stackrel{\varphi}{\longrightarrow} GL(V)##. We say that the elements if ##G## act (operate) on ##V\, : \, g.v = \varphi(g)(v)##.

A linear representation of a Lie algebra ##\mathfrak{g}## on a vector space ##V## is a Lie algebra homomorphism ##\mathfrak{g} \stackrel{\varphi}{\longrightarrow} \mathfrak{gl}(V)##. We say that the elements if ##\mathfrak{g}## act (operate) on ##V\, : \, g.v = \varphi(g)(v)##.