Determine the complete spectra and multiplicities of the shuffle multiplication matrices

Determine explicitly the eigenvalues and their multiplicities for the matrices of left multiplication by the shuffle elements \(\mathscr{Y}_{n,i}\) in \(\mathbb{C}\mathfrak{S}_n\), and for the corresponding \(q\)-deformed matrices \(\mathscr{M}^q_{n,i}\) in the generic Iwahori–Hecke algebra \(H_n(q)\), for all admissible \(n\) and \(i\).

Background

For each $1\leq i\leq \binom{n}{2}$, the paper defines $\mathscr{M}^q_{n,i}$ as the matrix of left multiplication by the length-i shuffle element $\mathscr{Y}_{n,i}$ in the standard basis of $H_n(q)$, and writes $\mathscr{M}_{n,i}$ for its specialization at $q=1$. The paper proves that $\mathscr{M}^q_{n,i}$ is $q$-symmetric and hence that $\mathscr{M}_{n,i}$ is symmetric, which guarantees diagonalizability in the latter case.