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.

About Pure mathematics seminars

We present regular seminars on a range of pure mathematics interests. Students, staff and visitors to UQ are welcome to attend, and to suggest speakers and topics.

Seminars are usually held on Tuesdays from 3pm to 3.50pm.

Talks comprise 45 minutes of speaking time plus five minutes for questions and discussion.

Information for speakers

Researchers in all pure mathematics fields attend our seminars, so please aim your presentation at a general mathematical audience.

Contact us

To volunteer to talk or to suggest a speaker, email Ole Warnaar or Travis Scrimshaw.


Goddard Building (08)