what is the conjugate of irreducible character of $G\wr S_n $?

Question

Assume $G$ is any finite group and field as a complex field.
The index set of irreducible representations of $G\wr S_n$ is set of all $k$-tuble of partitions $\widetilde{\lambda}=(\lambda_1,\lambda_2,\cdots \lambda_k)$ such that $\displaystyle \sum_{i=1}^k |\lambda_i|=n,$ where $k$ is number of conjugacy classes of $G$.
Suppose $\chi_\widetilde{\lambda}$ is the irreducible character of $G\wr S_n.$ Then what is the conjugate of $\chi_\widetilde{\lambda}?.$

That is, If $\bar{\chi}_\widetilde{\lambda_1}=\chi_\widetilde{\lambda_2}$, then what is the relation between $\widetilde{\lambda_1}$ and $\widetilde{\lambda_2}?.$

Answer 1

For a natural bijection between the irreducible characters of the wreath product and the set of $k$-tuples of partitions, you need to fix an ordering $\chi_1,\dots,\chi_k$ of the irreducible characters of $G$. Then taking the complex conjugate of a character of the wreath product corresponds to permuting the partitions in the corresponding $k$-tuple according to how complex conjugation permutes $\chi_1,\dots,\chi_k$. So if $\tilde{\lambda}$ and $\tilde{\mu}$ are the $k$-tuples corresponding to a character and its complex conjugate, then $\mu_i=\lambda_j$ for $\overline{\chi_i}=\chi_j$.