Relations between the higher Hamiltonians of the trigonometric and the rational spin Calogero-Sutherland models

In this paper we study the relations between the Hamiltonian hierarchy generated by trigonometric Cherednik-Dunkl operators and the one generated by rational Dunkl operators. We develop two nested structures relating these two hierarchies. The first nested relation reconstructs the higher rational spin Calogero-Sutherland Hamiltonians exactly from the trigonometric ones. The second nested relation reconstructs the higher trigonometric spin Calogero-Sutherland Hamiltonians as leading terms from the rational ones. In addition, we provide an explanation of the eigenvalues of the trigonometric spin Calogero-Sutherland Hamiltonians in terms of $N$-colored Young diagrams and Maya diagrams.