# The q-deformed Haldane--Shastry spin chain: from the affine Hecke algebra, via 'freezing', to exact eigenvectors

The Haldane--Shastry model is an exactly-solvable long-range spin chain whose remarkable properties include Yangian symmetry already at finite system size. Its q-deformation was found by D. Uglov in '95 in an e-print that seems to have gone by unnoticed. Last year I managed to express Uglov's Hamiltonian in a more friendly, pairwise form.

I will outline how this Hamiltonian is obtained by 'freezing' the spin generalisation of the Ruijsenaars--Schneider model (Macdonald operators). The generalised model enjoys quantum-affine symmetries that, by construction, are inherited by the spin chain upon freezing.

Our main result is an exact expression in closed form for the (highest-weight) eigenvectors at finite size, involving the symmetric square of the q-Vandermonde times a Macdonald polynomial (with p = q²) 'dressed' by the polynomial action of the Hecke algebra.

This is ongoing work together with V. Pasquier and D. Serban.

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