  We use the Darboux coordinate representation found by two of the authors
(L.Ch. and M.Sh.) for entries of general symplectic leaves of the $\mathcal
A_n$-groupoid of upper-triangular matrices to express roots of the
characteristic equation $\det(\mathbb A-\lambda \mathbb A^{\text{T}})=0$, with
$\mathbb A\in \mathcal A_n$, in terms of Casimirs of this Darboux coordinate
representation, which is based on cluster variables of Fock--Goncharov higher
Teichm\"uller spaces for the algebra $sl_n$. We show that roots of the
characteristic equation are simple monomials of cluster Casimir elements. This
statement remains valid in the quantum case as well. We consider a
generalization of $\mathcal A_n$-groupoid to a $\mathcal A_{Sp_{2m}}$-groupoid.
