178 research outputs found
From Jack to Double Jack Polynomials via the Supersymmetric Bridge
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 algebra
Jack superpolynomials: physical and combinatorial definitions
Jack superpolynomials are eigenfunctions of the supersymmetric extension of
the quantum trigonometric Calogero-Moser-Sutherland. They are orthogonal with
respect to the scalar product, dubbed physical, that is naturally induced by
this quantum-mechanical problem. But Jack superpolynomials can also be defined
more combinatorially, starting from the multiplicative bases of symmetric
superpolynomials, enforcing orthogonality with respect to a one-parameter
deformation of the combinatorial scalar product. Both constructions turns out
to be equivalent. This provides strong support for the correctness of the
various underlying constructions and for the pivotal role of Jack
superpolynomials in the theory of symmetric superpolynomials.Comment: 6 pages. To appear in the proceedings of the {\it XIII International
Colloquium on Integrable Systems and Quantum Groups}, Czech. J . Phys., June
17-19 2004, Doppler Institute, Czech Technical Universit
Jack polynomials in superspace
This work initiates the study of {\it orthogonal} symmetric polynomials in
superspace. Here we present two approaches leading to a family of orthogonal
polynomials in superspace that generalize the Jack polynomials. The first
approach relies on previous work by the authors in which eigenfunctions of the
supersymmetric extension of the trigonometric Calogero-Moser-Sutherland
Hamiltonian were constructed. Orthogonal eigenfunctions are now obtained by
diagonalizing the first nontrivial element of a bosonic tower of commuting
conserved charges not containing this Hamiltonian. Quite remarkably, the
expansion coefficients of these orthogonal eigenfunctions in the supermonomial
basis are stable with respect to the number of variables. The second and more
direct approach amounts to symmetrize products of non-symmetric Jack
polynomials with monomials in the fermionic variables. This time, the
orthogonality is inherited from the orthogonality of the non-symmetric Jack
polynomials, and the value of the norm is given explicitly.Comment: 28 pages. Corrected version of lemme 3 and other minor corrections
and 2 new references; version to appear in Commun. Math. Phy
Explicit formulas for the generalized Hermite polynomials in superspace
We provide explicit formulas for the orthogonal eigenfunctions of the
supersymmetric extension of the rational Calogero-Moser-Sutherland model with
harmonic confinement, i.e., the generalized Hermite (or Hi-Jack) polynomials in
superspace. The construction relies on the triangular action of the Hamiltonian
on the supermonomial basis. This translates into determinantal expressions for
the Hamiltonian's eigenfunctions.Comment: 19 pages. This is a recasting of the second part of the first version
of hep-th/0305038 which has been splitted in two articles. In this revised
version, the introduction has been rewritten and a new appendix has been
added. To appear in JP
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