# Conjugate representations in the representation theory of the symmetric groups

I first thought about posting this on physics stackexchange, but I think the chance of getting an answer is higher here.

Let $\alpha$ be a unitary irrep of the symmetric group $S_N$ and $U^{[\alpha]}$ the representation matrix in some orthonormal basis. Then the book "Ruben Pauncz: The construction of spin eigenfunctions - An exercise book" (it's a Quantum Chemistry book) writes in section 6.6 on p. 77 that the representation matrix of the conjugate representation $\alpha'$ is given by

$$U^{[\alpha']}(\sigma) = \text{sgn}(\sigma) \cdot U^{[\alpha]}(\sigma) \quad \forall\sigma\in S_N.$$

It is then shown that this "conjugate representation" is exactly the one that corresponds to mirroring the Young frame (exchanging rows and columns). For example, according to this definition, the conjugate of the trivial representation is the sign representation.

My question is the following: Has this definition anything to do with what is usually meant by the term "(complex) conjugate representation" (https://en.wikipedia.org/wiki/Complex_conjugate_representation, https://en.wikipedia.org/wiki/Dual_representation)? Since all irreps of $S_N$ can be written as matrices with real entries, they are self-conjugate (meaning that each irrep is its own conjugate). Is this simply a case of using the same term for two different things, or is there some connection that escaped my attention?

## 1 Answer

Irreducible representations of the symmetric group are labelled by partitions. When you reflect the Young diagram of a partition over the main diagonal, the resulting partition is commonly called the conjugate partition. This is the meaning of conjugate in this context.