I’ve been working through some of the theory of the representation theory of finite groups, and still cannot really work out a sensible answer to a question posed to me by Josh Lam about a year ago: why is the number of irreducible representations equal to the number of conjugacy classes?

Reason 1: “Mess around with characters”. Use some tricks involving Schur’s lemma to transfer info about characters to info about endomorphisms, and show that the characters of the irreducibles are an orthonormal system inside the vector space of all class functions. Then use similar tricks and the regular representation, to prove that the only class function orthogonal too all of them is the zero function. Hence the irreducible characters are a basis for the space of class functions, so done by dimension counting.

Reason 2: “Mess around with modules over general algebras”. A theory due to Wedderburn tells us that every semisimple module over an algebra can be decomposed into a direct sum of matrix algebras over a division algebra. This gives us a decomposition of the group algebra into simple matrix algebras

$\mathbb{C}G = \oplus_{i=1}^k M_k.$

Each summand corresponds to an irreducible representation, and since the centre of a matrix algebra is just the 1-dimensional subspace of scalar matrices, and the centre of the group algebra is generated by the class sums, taking the centre of both sides and counting dimensions gives the desired result.

Though these both shed different lights on the subject, I still can’t help but feel that there must be some form of duality or bigger theory which actually gives us a direct reason for these two objects to correspond. More broadly, the whole machinery of characters which derives from these kinds of deepish argument feels slightly incomplete, like a calculational trick adopted at the infancy of the subject which has then stuck as the subject has deepened where a more intuitive and natural set of tools should be available and able to make the initial setup of the subject more transparent than it is at the moment. Is anyone out there aware of any such understanding?