Jack Superpolynomials with Negative Fractional Parameter: Clustering Properties and Super-Virasoro Ideals

Abstract

Abstract. The Jack polynomials P is invariant under the action of certain differential operators which include half the Virasoro algebra. Moreover, the Jack polynomials in I (k,r) N admit clusters of size at most k: they vanish when k + 1 of their variables are identified, and they do not vanish when only k of them are identified. We generalize most of these properties to superspace using orthogonal eigenfunctions of the supersymmetric extension of the trigonometric Calogero-Moser-Sutherland model known as Jack superpolynomials. In particular, we show that the Jack superpolynomials P is stable with respect to the action of the negative-half of the super-Virasoro algebra. In addition, we show that the Jack superpolynomials in I (k,r) N vanish when k + 1 of their commuting variables are equal, and conjecture that they do not vanish when only k of them are identified. This allows us to conclude that the standard Jack polynomials with prescribed symmetry should satisfy similar clustering properties. Finally, we conjecture that the elements of I (k,2) N provide a basis for the subspace of symmetric superpolynomials in N variables that vanish when k + 1 commuting variables are set equal to each other.

Supersymmetric Ruijsenaars-Schneider Model

“…[17]. Recently, the sJacks were used in superconformal field theory [18] and were shown to possess clustering properties similar to those observed for the quantum fractional Hall states [19].…”

Pieri Rules for the Jack Polynomials in Superspace and the 6-Vertex Model

“…with N the number of variables (which will soon become irrelevant). The explicit action of Q and q ⊥ , as well as ofẽ 0 , on Jack polynomials in superspace was obtained in [11] e 0 P…”

The super-Virasoro singular vectors and Jack superpolynomials relationship revisited

“…The projective limit Σ = lim ← − Σ N (A. 10) corresponds to the case where we have infinitely many indeterminates z 1 , θ 1 , z 2 , θ 2 , . .…”

“…Most of the properties of Jack polynomials can be lifted to the super-case [6,8,10]. Here we present those that are used in this work.…”

“…The superpolynomials that vanish when three (bosonic) variables coincide have been studied in [10]. Let us denote their vector space by F (2) l , that is,…”