Basics of group representation theoryJuly 10, 2009

Posted by Akhil Mathew in algebra, representation theory.
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Start with a group ${G}$. At least for now, we’re essentially going to be constructed with finite groups, but many of these constructions generalize. A representation of ${G}$ is essentially an action of ${G}$ on a finite-dimensional complex vector space ${V}$.
Definition 1 A representation of the group ${G}$ is a finite-dimensional complex vector space ${V}$ and a group-homomorphism ${G \rightarrow Aut(G)}$. In other words, it is a group homomorphism ${G \rightarrow GL_n(V)}$, where ${n = \dim \ V}$, and ${GL_n}$ is the group of invertible ${n}$-by-${n}$ matrices.
An easy example is just the unit representation, sending each ${g \in G}$ to the identity matrix. (more…)