group representation

Consider a group \(G\) and a vector space V. Consider the general linear group (the group of mappings from \(V\) to itself).

We can make a group homomorphism from \(G\) to \(GL(V)\). Then, we can describe \(G\) with \(GL(V)\). Then, we say that we are representing \(G\) with \(GL(V)\). Often, you will see the language "a representation \(\rho\) is a homomorphsim \(\rho: G \rightarrow GL(V)\)"

Consider two group representations \(\pi: G \rightarrow GL(V)\) and \(\pi': G \rightarrow GL(W)\) if there exists a mapping \(A: V \rightarrow W\) such that for every \(v \in V\) and \(g \in G\), \(A \pi(g) v = \pi'(g) A v\). That is

• \(A\) is a similarity transform (see matrix similarity)
• \(A\) preserves the group action
• \(A\) is \(G\) -equivariant (see group invariant and group equivariant functions)

Why is it important to consider group isomorphisms? Because any character of \(\pi\), such as irreducibility, number of irreducible constituents, character, carry over with the isomorphism. (See stack overflow answer)

• If we choose a basis, we can make a homomorphism from group elements to \(GL(n,F)\), the set of \(n\times n\) matrices (see wikipedia)
• remember that matrices take coordinates as input. And coordinates are given with respect to a basis.

• two representations of a lie group \(G\) on vector spaces \(V\) and \(W\) are equivalent if they have the same matrix representations with respect to a choice of basis for \(V\) and \(W\) (from lie group representation).
• intuition: consider the case when \(V=W\), then any choice of basis defines a different coordinate system. The same representation, adjusted for different coordinate systems, are all doing the same thing.
• Also consider a change of basis \(U\) and matrix representations of group elements \(A\) and \(B\). Then notice the fact that \(UAU^{-1}UBU^{-1} = UABU^{-1}\), i.e., the matrix \(AB\) after similarity transform

• wikipedia