representation

A representation turns an abstract group into concrete matrices, so you can study a possibly mysterious group with the linear algebra you already command. The idea: have each group element ACT as an invertible matrix, in a way that respects the group's multiplication. Symmetries you cannot picture become matrices you can compute with.

Formally a representation of a group G on a vector space V is a homomorphism rho : G -> GL(V), meaning rho(g h) = rho(g) rho(h) and rho(e) = I. Each element g is realized as the invertible operator rho(g), and composing in the group matches multiplying the operators. The dimension of V is the dimension of the representation; V is called the representation space.

The point is that abstract groups become LINEAR symmetries, where eigenvalues, traces, and invariant subspaces are available. A subspace W <= V fixed by every rho(g) is a subrepresentation; a representation with no nontrivial invariant subspace is irreducible — these are the atoms from which all representations are assembled, the heart of representation theory.

Two caveats. A representation need not be faithful: if rho has a kernel, different group elements collapse to the same matrix and information is lost (the trivial representation g -> 1 is the extreme case). And for a Lie group, differentiating rho at the identity yields a representation of its Lie algebra — so group and algebra representations come in matched pairs.