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.
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 2 to 3pm.
Talks comprise 45 minutes of speaking time plus five minutes for questions and discussion.
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