# Identity involving partitions coming from representations of alternating groups

It is not difficult to show that the number of conjugacy classes in the alternating group $A_n$ is given by

classes in the alternating group = no. of even partitions + no. of self-transpose partitions

Note that a partition is even if it is the cycle decomposition of an even permutation.

Likewise, using some Clifford theory and the representation theory of $S_n$, one can show that the number of irreducible representations of $A_n$ is given by

irreps. of $A_n$ = $\frac 12$(no. of non-self transpose partitions) + no. of self-transpose partitions

Equating these leads to the identity:

$2\times$ no. of even partitions - no. of self-transpose partitions = no. of partitions

In his book Representations of Finite Groups, Musili refers to this as a bizarre identity.

Question. Is there a proof of this identity which does not use the representation theory of alternating groups? Better still, is there a bijective proof?

• What is a symmetric partition? – Sasha Oct 23 '12 at 9:39
• I meant to say a self-transpose one. I edited my question. – Amritanshu Prasad Oct 23 '12 at 10:42
• See Marc's answer here math.stackexchange.com/questions/102242/… – Gjergji Zaimi Oct 23 '12 at 12:10
• See also Exercise 1.22(b) of Enumerative Combinatorics, vol. 1, 2nd ed., and the ends of Sections 7.7 and 7.14 of vol. 2. – Richard Stanley Oct 23 '12 at 15:27
• Thanks Gjergji Zaimi; Marc's answer is in fact a bijective proof. – Amritanshu Prasad Oct 25 '12 at 4:28

A very simple explanation for this identity comes from the theory of symmetric functions. The ring $\Lambda$ of symmetric functions in infinitely many variables comes with an involution $\omega$, which interchanges the complete symmetric function $h_\lambda$ with the elementary symmetric function $e_\lambda$ for each partition $\lambda$.
Comparing the answers obtained for the trace of $\omega$ on homogeneous symmetric functions of degree $n$ using Schur functions and power sum symmetric functions yields the identity in question, for $\omega(s_\lambda)=s_{\lambda'}$ (giving trace as the number of self-transpose partitions) and $\omega(p_\lambda)=\epsilon(\lambda)p_\lambda$, where $\epsilon(\lambda)$ is the sign of a permutation with cycle decomposition $\lambda$ (giving trace as number of even partitions minus number of odd partitions).