Generalised discriminants, deformed Calogero–Moser–Sutherland operators and super-Jack polynomials

Abstract

It is shown that the deformed Calogero-Moser-Sutherland (CMS) operators can be described as the restrictions on certain affine subvarieties (called generalised discriminants) of the usual CMS operators for infinite number of particles. The ideals of these varieties are shown to be generated by the Jack symmetric functions related to the Young diagrams with special geometry. A general structure of the ideals which are invariant under the action of the quantum CMS integrals is discussed in this context. The shifted super-Jack polynomials are introduced and combinatorial formulas for them and for super-Jack polynomials are given. r 2004 Elsevier Inc. All rights reserved.

Painlevé IV and degenerate Gaussian unitary ensembles

“…This type of generalisation is integrable in the case of two types of particles [6], [24]. More remarkably, Sergeev and Veselov showed that the corresponding quantum Hamiltonian can be obtained by applying a restriction procedure on the Calogero-Moser-Sutherland Hamiltonian in the infinite dimensional space to the appropriate discriminant [25]. We plan to elaborate these relations in future.…”

Deformed Quantum Calogero-Moser Problems and Lie Superalgebras

“…The proof is effective but not conceptual. We present more conceptual proof for A(n, m) series in our paper [38], where the algebraic varieties corresponding to the rings Λ 0 R,B and relations with the theory of Jack polynomials are also discussed in more detail.…”

“…The relation with the theory of Jack polynomials is actually much deeper. We discuss this in detail in our paper [38] (see also [13], [14]). …”

Collective field formulation of the multi-species Calogero model and its duality symmetries

“…The Fourier transforms a c (k) of these operators, and their hermtian conjugates, defined as in (40) and (43), obey the commutation relations…”

“…The authors of [39] introduced deformed Calogero models, related to root systems of superalgebras, and gave effectively a proof of their integrability. In [40] they presented a more conceptual proof by using shifted super-Jack polynomials. In related developments, the authors of [41,42] introduced a supersymmetric generalizations of the CM which was based on Jacobians for the radial coordinates on certain superspaces.…”