Distinctness of the wreath-matrix eigenvalues

Determine whether the eigenvalues \(\lambda_1,\lambda_2,\ldots,\lambda_k\) of the wreath matrix \(M(n,k)\) described in Theorem 1 are all distinct.

Background

The paper derives all eigenvalues of the wreath matrix and their stated multiplicities, but notes that the eigenvalues listed in the main theorem need not be pairwise distinct. If eigenvalues coincide, their eigenspace multiplicities must be combined.

The authors identify the distinctness question as a specific property of the wreath matrix for further investigation. It is posed after the main spectral results and is not resolved in the paper.