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

The quantum relativistic Toda Hamiltonians for classical root systems are obtained as the q-Whittaker limit of (dual) Macdonald operators or specialized Koornwinder operators. They are the conserved quantities of discrete evolutions in a family of quantum cluster algebras known as Q-systems. The polynomial eigenfunctions (q-Whittaker functions) can be constructed from the action of a subset of A-type cluster variables, which act as raising operators. This gives a uniform description for all classical root systems of such eigenfuctions. In some cases, the augmented cluster algebra quiver also gives a candidate for Baxter operators commuting with the quantum Hamiltonians. Joint with Philippe Di Francesco.