The square of the Vandermonde determinant and itsq-generalization

Laughlin's function on a cylinder: plasma analogy and representation as a quantum polymer

“…The coefficients a N (m) are specific to the cylinder geometry and have been analyzed by Rezayi and Haldane [15], while the integers b N (m) play a role as well for the usual "Laughlin droplet" geometry and have inspired a certain amount of combinatorial research [11,12,21,22]. It is practical to associate with each finite sequence m a sequence of 1's and 0's characterizing occupancy or non-occupancy of lattice sites {−1, 0, 1, ..., 3N − 2}.…”

Macdonald polynomials at t=qk

“…This result is interesting in the context of the fractional quantum Hall effect [8], since it implies properties of the expansion of the powers of the discriminant in the Schur basis [3,6,14]. It implies also that the Macdonald polynomials (at t = q k ) are characterized by the eigenvalues of an operator M (described in terms of isobaric divided differences) whose eigenspaces are of dimension 1.…”

“…This deformation is also relevant in the study of the expansion of i< j (x i − x j ) 2k in the Schur basis (for example, we generalized in [2] a result of [6]). …”

Hankel hyperdeterminants, rectangular Jack polynomials and even powers of the Vandermonde

“…In particular it is of considerable interest to determine for which partitions the coefficients of the Schur functions in the expansion of the square of Vandermonde vanishes [5,16,27,31,32]. The aim of this subsection is to give a hyperdeterminantal expression for the coefficient of S λ (X) in (X) 2k .…”