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.



An $n!\times n!$ determinant
This determinant came up, and was evaluated, in the [comments of the Secret Blogging Seminar](http://sbseminar.wordpress.com/2009/02/25/a-warmup-gl_t-for-t-not-an-integer/#comment-4870). The motivation there was that it vanishes if and only if $V^{\otimes n}$ has neglible endomorphisms in Deligne's category of "$GL\_x$ representations for noninteger $x$". Here $V$ is the "$x$-dimensional representation of $GL\_x$". See that post for more.


