TITLE: An $n!\times n!$ determinant
QUESTION [19 upvotes]: Let us consider the matrix $A$ with its rows and columns enumerated by the elements of $S_n$ with $A_{\sigma\tau}=x^{c(\sigma\tau^{-1})}$ where $c()$ is the number of cycles in a permutation's decomposition. I'm interested in $|A|$. More specifically I aim to prove that all of its roots as of a polynomial in $x$ are integers between $-n+1$ and $n-1$ but the roots' multiplicities would also be nice to know.

REPLY [9 votes]: Here (Darij Grinberg, A representation-theoretical solution to MathOverflow question #88399) is the proof that I hinted at in the comments section in more details. Repeated mistakes absorbed most of the time I spent writing it, which is why it took four days; let me apologize for this.