regular representation

Groups act on vector spaces via their group representation (see group action). But groups can also as a vector space. Let us denote the vector space of any set (not just a group) \(X\) as: \[\mathbb{C}X = \left\{ \sum_{x\in X} c_x x \mid c_x \in \mathbb{C} \right\}\]

Then, the (left) regular representation of \(G\) is the homomorphism \(\rho: G \rightarrow GL(\mathbb{C}G)\) given by: \[\rho_g \sum_{h\in G} c_h h = \sum_{h\in G} c_h gh = \sum_{x\in G} c_{g^{-1} x} x\] where \(x=gh\) and \(g\in G\)

A couple things to note:

• remember that \(\mathbb{C}G\) is a vector space, so the homomorphism takes elements of \(G\) to matrices
• \(\sum_{h\in G} c_h h\) is a vector in that vector space
• \(c_{g^{-1}x} = c_h\)