Equivalence of definitions for regular representations

[Kreizhn]

There seem to be two definitions for a regular representation of a group, with respect to a field k. In particular, one definition is that the regular representation is just left multiplication on the group algebra kG, while the other is defined on the set of all functions f: G \to k. I do not see why these are equivalent, and would appreciate any advice as to why this is the case.

 

[morphism]

(I'm assuming G is a finite group.) The element $\sum_{g \in G} c_g g$ in kG can be thought of as the function $G \to k$ defined by $g \mapsto c_g$. Conversely, a function $f \colon G \to k$ gives rise to the element $\sum_g f(g) g \in kG$. From this it's easy to see that the two vector spaces kG and {functions $G \to K$} are isomorphic; in fact the map $\sum_g c_g g \mapsto (g \mapsto c_g)$ is an isomorphism.

Now all you have to do is check that this isomorphism respects the G-action. You've already indicated that the action on kG is given by left multiplication. The action of G on a function $f \colon G \to k$ is defined by $(h \cdot f)(g) = f(h^{-1}g)$ (for $h \in G$). Now note that
$$ h \sum_g c_g g = \sum_g c_g hg = \sum_{h^{-1}g} c_{h^{-1}g} g. $$ This shows that the isomorphism is G-linear.

 

[Kreizhn]

Excellent, thank you.