Characteristic equation for symplectic groupoid and cluster algebras

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 Aₙ-groupoid of upper-triangular matrices to express roots of the characteristic equation (A-λ A^T) =0, with A Aₙ, in terms of Casimirs of this Darboux coordinate representation, which is based on cluster variables of Fock--Goncharov higher Teichmüller spaces for the algebra slₙ. 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 Aₙ-groupoid to a Aₒ₏_₂₌-groupoid.