The Characters of the Symmetric Group
Francis D. Murnaghan · Proceedings of the National Academy of Sciences · 1971
A short and simple derivation of the formula of Frobenius, which gives the dimensions of the irreducible representations of S(n), the symmetric group on any number, n, of symbols, is given. These dimensions are the characters of the identity element of the group, i.e., of the element all of whose cycles are unary. It is shown how a slight modification of Frobenius' formula yields, when n = 2p is even, the characters of an element of S(n) all of whose cycles are binary and, when n = 3p is a multiple of 3, the characters of an element of S(n) all of whose cycles are ternary and, generally, when n = kp is a multiple of any positive integer k, the characters of an element of S(n) all of whose cycles are of length k. It is noteworthy that the calculations become simpler, rather than more complicated, as k increases. Finally, this paper shows how to derive from Frobenius' formula the characters of an element of S(n) which has at least one unary cycle and, from the present modifications of Frobenius' formula, the characters of an element of S(n) which has at least one cycle of length k, k = 2, 3,..., n.