Luc Lapointe

Tableau atoms and a new Macdonald positivity conjecture

A snap action fluid control valve, the operation of which is controlled by a relatively slow acting thermally responsive actuator member. The valve of this invention is particularly adapted for use in controlling flow of fluid to a fluid operable clutch or the like for operation thereof. The thermally responsive actuator portion of the valve senses temperature of a fluid, the temperature of which is responsive to operation of a fan which is operated through the clutch.

Tableaux on k + 1-crores, reduced words for affine permutations, and k -Schur expansions

The k-Young lattice Yk is a partial order on partitons with no part larger than k. This weak subposet of the Young lattice originated (Duke Math. J. 116 (2003) 103-146) from the study of the k-Schur functions sλ(k), Symmetric functions that from a natural basis of the space spanned by homogeneous funtions indexed by k-bounded partitions. The chains in the k-Young lattice are induced by a Pieritype rule experimentally satisfied by the k-Schur functions. Here, using a natural bijection between k-bounded partitons and k + 1-cores, we establish an algorithm for identifying chains in the k- Young lattice with certain tableaux on k + 1 cores. This algorithm reveals that the k-Young lattice is isomorphic to the weak order on the quotient of the affine symmetric gruop S˜k+1 by a maximal parabolic subgruop. From this, the conjectured k-Pieri rule implies that the k-Kostka matrix connecting the homogeneous basis {hλ}λ∈Yk to {sλ(k)}λ∈Yk may now be obtained by counting appropriate classes of tableaux on k + 1-cores. This suggests that the conjecturally positive k-Schur expansion coefficients for Macdonald polynomials (reducing to q, t-Kostka polynomials for large k) could be described by a q, t-statistic on these tableaux, or equivalently on reduced words for affine permutations.

QUANTUM COHOMOLOGY AND THE k-SCHUR BASIS

We prove that structure constants related to Hecke algebras at roots of unity are special cases of k-Littlewood-Richardson coefficients associated to a product of k-Schur functions. As a consequence, both the 3-point Gromov-Witten invariants appearing in the quantum cohomology of the Grassmannian, and the fusion coefficients for the WZW conformal field theories associated to su(l) are shown to be k-Littlewood-Richardson coefficients. From this, Mark Shimozono conjectured that the k-Schur functions form the Schubert basis for the homology of the loop Grassmannian, whereas k-Schur coproducts correspond to the integral cohomology of the loop Grassmannian. We introduce dual k-Schur functions defined on weights of k-tableaux that, given Shimozonos conjecture, form the Schubert basis for the cohomology of the loop Grassmannian. We derive several properties of these functions that extend those of skew Schur functions.

Supersymmetric Calogero–Moser–Sutherland models and Jack superpolynomials

Abstract A new generalization of the Jack polynomials that incorporates fermionic variables is presented. These Jack superpolynomials are constructed as those eigenfunctions of the supersymmetric extension of the trigonometric Calogero–Moser–Sutherland (CMS) model that decomposes triangularly in terms of the symmetric monomial superfunctions. Many explicit examples are displayed. Furthermore, various new results have been obtained for the supersymmetric version of the CMS models: the Lax formulation, the construction of the Dunkl operators and the explicit expressions for the conserved charges. The reformulation of the models in terms of the exchange-operator formalism is a crucial aspect of our analysis.

Schur function analogs for a filtration of the symmetric function space

We work here with the linear span Λt(k) of Hall-Littlewood polynomials indexed by partitions whose first part is no larger than k. The sequence of spaces Λt(k) yields a filtration of the space Λ of symmetric functions in an infinite alphabet X. In joint work with Lascoux [4] we gave a combinatorial construction of a family of symmetric polynomials {Aλ(k)[X; t]}λ1 ≤ k, with N[t]-integral Schur function expansions, which we conjectured to yield a basis for Λt(k). Our primary motivation for this construction is to provide a positive integral factorization of the Macdonald q, t-Kostka matrix. More precisely, we conjecture that the connection coefficients expressing the Hall-Littlewood or Macdonald polynomials belonging to Λt(k) in terms of the basis {Aλ(k)[X; t]}λ1 ≤ k are polynomials in N [q, t]. We give here a purely algebraic construction of a new family {sλ(k)[X; t]}λ1 ≤ kof polynomials in Λt(k) which we conjecture is identical to {Aλ(k)[X; t]}λ1 ≤ k. We prove that {sλ(k) [X; t]}λ1 ≤ k is in fact a basis of Λt(k) and derive several further properties including that sλ(k)[X; t] reduces to the Schur function sλ[X] for sufficiently large k. We also state a number of conjectures which reveal that the polynomials {sλ(k)[X; t]}λ1 ≤ k are in fact the natural analogues of Schur functions for the space Λt(k).

Jack Polynomials in Superspace

AbstractThis work initiates the study of 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.

A note on (p, q)-oscillators and bibasic hypergeometric functions

The authors examine the relation between the representation theory of a two-parameter deformation of the oscillator algebra and certain bibasic Laguerre functions and polynomials.

Jack superpolynomials, superpartition ordering and determinantal formulas

Abstract: We call superpartitions the indices of the eigenfunctions of the supersymmetric extension of the trigonometric Calogero-Moser-Sutherland model. We obtain an ordering on superpartitions from the explicit action of the models Hamiltonian on monomial superfunctions. This allows to define Jack superpolynomials as the unique eigenfunctions of the model that decompose triangularly, with respect to this ordering, on the basis of monomial superfunctions. This further leads to a simple and explicit determinantal expression for the Jack superpolynomials.

Classical symmetric functions in superspace

We present the basic elements of a generalization of symmetric function theory involving functions of commuting and anticommuting (Grassmannian) variables. These new functions, called symmetric functions in superspace, are invariant under the diagonal action of the symmetric group on the sets of commuting and anticommuting variables. In this work, we present the superspace extension of the classical bases, namely, the monomial symmetric functions, the elementary symmetric functions, the completely symmetric functions, and the power sums. Various basic results, such as the generating functions for the multiplicative bases, Cauchy formulas, involution operations as well as the combinatorial scalar product are also generalized.

Orthogonality of Jack polynomials in superspace

Abstract Jack polynomials in superspace, orthogonal with respect to a “combinatorial” scalar product, are constructed. They are shown to coincide with the Jack polynomials in superspace, orthogonal with respect to an “analytical” scalar product, introduced in [P. Desrosiers, L. Lapointe, P. Mathieu, Jack polynomials in superspace, Comm. Math. Phys. 242 (2003) 331–360] as eigenfunctions of a supersymmetric quantum mechanical many-body problem. The results of this article rely on generalizing (to include an extra parameter) the theory of classical symmetric functions in superspace developed recently in [P. Desrosiers, L. Lapointe, P. Mathieu, Classical symmetric functions in superspace, J. Algebraic Combin. 24 (2006) 209–238].