Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 051, 15 pages      arXiv:1503.09029      https://doi.org/10.3842/SIGMA.2015.051
Contribution to the Special Issue on Exact Solvability and Symmetry Avatars in honour of Luc Vinet

From Jack to Double Jack Polynomials via the Supersymmetric Bridge

Luc Lapointe a and Pierre Mathieu b
a) Instituto de Matemática y Física, Universidad de Talca, 2 norte 685, Talca, Chile
b) Département de physique, de génie physique et d'optique, Université Laval, Québec, Canada, G1V 0A6

Received March 31, 2015, in final form June 25, 2015; Published online July 02, 2015

Abstract
The Calogero-Sutherland model occurs in a large number of physical contexts, either directly or via its eigenfunctions, the Jack polynomials. The supersymmetric counterpart of this model, although much less ubiquitous, has an equally rich structure. In particular, its eigenfunctions, the Jack superpolynomials, appear to share the very same remarkable combinatorial and structural properties as their non-supersymmetric version. These super-functions are parametrized by superpartitions with fixed bosonic and fermionic degrees. Now, a truly amazing feature pops out when the fermionic degree is sufficiently large: the Jack superpolynomials stabilize and factorize. Their stability is with respect to their expansion in terms of an elementary basis where, in the stable sector, the expansion coefficients become independent of the fermionic degree. Their factorization is seen when the fermionic variables are stripped off in a suitable way which results in a product of two ordinary Jack polynomials (somewhat modified by plethystic transformations), dubbed the double Jack polynomials. Here, in addition to spelling out these results, which were first obtained in the context of Macdonal superpolynomials, we provide a heuristic derivation of the Jack superpolynomial case by performing simple manipulations on the supersymmetric eigen-operators, rendering them independent of the number of particles and of the fermionic degree. In addition, we work out the expression of the Hamiltonian which characterizes the double Jacks. This Hamiltonian, which defines a new integrable system, involves not only the expected Calogero-Sutherland pieces but also combinations of the generators of an underlying affine ${\widehat{\mathfrak {sl}}_2}$ algebra.

Key words: Jack polynomials; supersymmetry; Calogero-Sutherland model; integrable quantum many-body problem; affine algebra.

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