Showing papers by "Pierre Mathieu published in 2015"

Supersymmetric Ruijsenaars-Schneider Model

Abstract: An integrable supersymmetric generalization of the trigonometric Ruijsenaars-Schneider model is presented whose symmetry algebra includes the super Poincar\'e algebra. Moreover, its Hamiltonian is shown to be diagonalized by the recently introduced Macdonald superpolynomials. Somewhat surprisingly, the consistency of the scalar product forces the discreteness of the Hilbert space.

Schur Superpolynomials: Combinatorial Definition and Pieri Rule

Abstract: Schur superpolynomials have been introduced recently as limiting cases of the Macdonald superpolynomials. It turns out that there are two natural super-extensions of the Schur polynomials: in the limit q = t = 0 and q = t!1, corresponding respectively to the Schur superpolynomials and their dual. However, a direct definition is missing. Here, we present a conjectural combinatorial definition for both of them, each being formulated in terms of a distinct extension of semi-standard tableaux. These two formulations are linked by another conjectural result, the Pieri rule for the Schur superpolynomials. Indeed, and this is an interesting novelty of the super case, the successive insertions of rows governed by this Pieri rule do not generate the tableaux underlying the Schur superpolynomials combinatorial construction, but rather those pertaining to their dual versions. As an aside, we present various extensions of the Schur bilinear identity.

Double Macdonald polynomials as the stable limit of Macdonald superpolynomials

Abstract: Macdonald superpolynomials provide a remarkably rich generalization of the usual Macdonald polynomials The starting point of this work is the observation of a previously unnoticed stability property of the Macdonald superpolynomials when the fermionic sector $$m$$ m is sufficiently large: their decomposition in the monomial basis is then independent of $$m$$ m These stable superpolynomials are readily mapped into bisymmetric polynomials, an operation that spoils the ring structure but drastically simplifies the associated vector space Our main result is a factorization of the (stable) bisymmetric Macdonald polynomials, called double Macdonald polynomials and indexed by pairs of partitions, into a product of Macdonald polynomials (albeit subject to non-trivial plethystic transformations) As an off-shoot, we note that, after multiplication by a $$t$$ t -Vandermonde determinant, this provides explicit formulas for a large class of Macdonald polynomials with prescribed symmetry The factorization of the double Macdonald polynomials leads immediately to the generalization of basically every elementary properties of the Macdonald polynomials to the double case (norm, kernel, duality, evaluation, positivity, etc) When lifted back to superspace, this validates various previously formulated conjectures in the stable regime The $$q,t$$ q , t -Kostka coefficients associated to the double Macdonald polynomials are shown to be $$q,t$$ q , t -analogs of the dimensions of the irreducible representations of the hyperoctahedral group $$B_n$$ B n Moreover, a Nabla operator on the double Macdonald polynomials is defined, and its action on a certain bisymmetric Schur function can be interpreted as the Frobenius series of a bigraded module of dimension $$(2n+1)^n$$ ( 2 n + 1 ) n , a formula again characteristic of the Coxeter group of type $$B_n$$ B n Finally, as a side result, we obtain a simple identity involving products of four Littlewood-Richardson coefficients

From Jack to Double Jack Polynomials via the Supersymmetric Bridge

Abstract: The Calogero{Sutherland model occurs in a large number of physical contexts, either directly or via its eigenfunctions, the Jack polynomials The supersymmetric coun- terpart 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 b sl2 algebra

N>=2 symmetric superpolynomials

Abstract: The theory of symmetric functions has been extended to the case where each variable is paired with an anticommuting one. The resulting expressions, dubbed superpolynomials, provide the natural N=1 supersymmetric version of the classical bases of symmetric functions. Here we consider the case where two independent anticommuting variables are attached to each ordinary variable. The N=2 super-version of the monomial, elementary, homogeneous symmetric functions, as well as the power sums, are then constructed systematically (using an exterior-differential formalism for the multiplicative bases), these functions being now indexed by a novel type of superpartitions. Moreover, the scalar product of power sums turns out to have a natural N=2 generalization which preserves the duality between the monomial and homogeneous bases. All these results are then generalized to an arbitrary value of N. Finally, for N=2, the scalar product and the homogenous functions are shown to have a one-parameter deformation, a result that prepares the ground for the yet-to-be-defined N=2 Jack superpolynomials.