The alternating group consists of the identity, the three double transpositions and the eight -cycles. Its conjugacy classes are found with the splitting criterion for conjugacy classes in (Proposition 11.4.3): a conjugacy class of contained in remains a single conjugacy class of if some odd permutation commutes with one of its elements, and otherwise it splits into two classes of equal size. The transposition commutes with , so the double transpositions form one class of . The centralizer of in has order (Proposition 3.2.13), so it is , and the eight -cycles split into the two classes So has four conjugacy classes, and therefore four irreducible representations, the squares of whose dimensions add up to .
The -dimensional representations. The subgroup of Section 14.1, consisting of and the three double transpositions, is a union of conjugacy classes, hence normal in (Proposition 3.2.13), and has order , so it is cyclic, generated by the coset of . Every homomorphism is trivial on : the three double transpositions are conjugate, so takes a common value on them, with ; and gives , so . So factors through , and with we obtain exactly three -dimensional representations , , , sending to , , respectively. (Here and are new; the alternating representation of becomes trivial on .) Since , their values on are , , .
The last representation. The fourth irreducible representation has dimension . The standard representation of , with acting through the inclusion , has character on , and so remains irreducible on . It is the fourth irreducible representation.
Note. The representation of does not stay irreducible on . Its character on is , which equals because ; so on , is isomorphic to . This also follows from the description of above: acts trivially, and maps onto the subgroup of , whose generator acts on the standard representation of with eigenvalues and (Lemma 13.3.4).