Appears in collection : Combinatorics and Arithmetic for Physics: special days 2023

One can associate with any finite poset with N elements a square matrix depending on 2(N − 1) variables, such that each matrix coefficient is just one of variables. Remarkably, all eigenvalues of this matrix are integer linear combinations of variables. Even in the case of the trivial poset, the structure of eigenvalues gives a new result concerning representations of the symmetric group. This is a joint work with R.Kenyon, O.Ogievetsky, A.Pohoata, W.Sawin and S.Shlosman.