Macdonald polynomials at $t=q^k$

Abstract

We investigate the homogeneous symmetric Macdonald polynomials P λ (X; q, t) for the specialization t = q k . We show an identity relying the polynomials P λ (X; q, q k ) and P λ 1−q 1−q k X; q, q k . As a consequence, we describe an operator whose eigenvalues characterize the polynomials P λ (X; q, q k ).

Hyperdeterminantal computation for the Laughlin wavefunction

“…One possible approach would consist in searching for the latter operator in the double affine Hecke algebra. Indeed, in previous articles, two of the authors gave q-deformations [3,18] which can be written as symmetric Macdonald functions indexed by rectangular or staircase partitions for some specializations of the parameters (which made us think that the Hecke algebra may play a role). We have not identified the operator yet.…”