I'm looking for a text I could cite that *explicitly* states the following result: for $\chi^\lambda$ the irreducible character of the symmetric group indexed by the partition $\lambda$, and for $\sigma \in \mathfrak{S}\_n$ and $\lambda \vdash n$,


$\chi^{\lambda'}(\sigma) = (-1)^{n-\ell(\sigma)} \chi^\lambda(\sigma),$


where $\lambda'$ is the dual partition, and $\ell(\sigma)$ is the length of the cycle type partition associated to $\sigma$.


It is not a hard result to prove (a simple combinatorial method, for instance, is to invoke the $\omega$ involution from Chapter 7 of Stanley's *Enumerative Combinatorics*). What I'm hoping for is a text that states this result directly (even if as an exercise) that could be cited without introducing background material and breaking the flow of a paper whose main topic is somewhat different from combinatorial representation theory.



Reference request: $\chi^{\lambda'}(\sigma) = (-1)^{n-\ell(\sigma)} \chi^\lambda(\sigma),$ for characters of the symmetric group
This is example 2 in page 116 of MacDonald's book, "Symmetric Functions and Hall Polynomials"


