These questions are motivated only by curiosity.
Take a Young tableau of shape $(\lambda_1,\lambda_2,\ldots,\lambda_n)$, where $\lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_n$. Is there any physical meaning or importance to the eigenvalues of such a tableau when embedded into a matrix (putting zeros outside the shape)? To be clear, I'm interested in combinatorial meanings. For example,
\begin{pmatrix} 1 & 2 & 5 & 6 \\ 3 & 4 & 8 & 0 \\ 7 & 10 & 0 & 0 \\ 9 & 0 & 0 & 0 \end{pmatrix}
has eigenvalues $\{-9.73,-5.43,5.62,14.53\}$.
Alternatively, instead of writing in the young tableau square values, write in the hook lengths of each square: \begin{pmatrix} 7 & 5 & 3 & 1 \\ 5 & 3 & 1 & 0 \\ 3 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{pmatrix}
The product of the hook lengths is intimately connected to the number of young tableaux of that shape. Is there a meaning to the eigenvalues of the resulting matrix? Obviously the determinant will be just 1 in this case, and 0 for anything non-staircase.
Hypothetically, the hook-length matrix (as well as the square value matrix) can be interpreted as an adjacency graph, so I can see the eigenvalues corresponding to something akin to the spectrum of the corresponding graph.