Searcharxiv⌕ Search

arXiv subjects

Luc Lapointe

Publications and source records attributed to Luc Lapointe.

At least 37 records · Page 2Linked to original sources
arXiv2013

Charge on tableaux and the poset of k-shapes

A poset on a certain class of partitions known as k-shapes was recently introduced to provide a combinatorial rule for the expansion of a (k-1)-Schur functions into k-Schur functions at t=1. The main ingredient in this construction was a bijection, which we call the weak bijection, that associates to a k-tableau a pair made out of a (k-1)-tableau and a path in the poset of k-shapes. We define here a concept of charge on k-tableaux (which conjecturally gives a combinatorial interpretation for the expansion coefficients of Hall-Littlewood polynomials into k-Schur functions), and show that it is compatible in the standard case with the weak bijection. In particular, we obtain that the usual charge of a standard tableau of size n is equal to the sum of the charges of its corresponding paths in the poset of k-shapes, for k=2,3...n.

math.CO↗
arXiv2012

Jack superpolynomials with negative fractional parameter: clustering properties and super-Virasoro ideals

The Jack polynomials P_λ^{(α)} at α=-(k+1)/(r-1) indexed by certain (k,r,N)-admissible partitions are known to span an ideal I^{(k,r)}_N of the space of symmetric functions in N variables. The ideal I^{(k,r)}_N is invariant under the action of certain differential operators which include half the Virasoro algebra. Moreover, the Jack polynomials in I^{(k,r)}_N admit clusters of size at most k: they vanish when k+1 of their variables are identified, and they do not vanish when only k of them are identified. We generalize most of these properties to superspace using orthogonal eigenfunctions of the supersymmetric extension of the trigonometric Calogero-Moser-Sutherland model known as Jack superpolynomials. In particular, we show that the Jack superpolynomials P_Λ^{(α)} at α=-(k+1)/(r-1) indexed by certain (k,r,N)-admissible superpartitions span an ideal {\mathcal I}^{(k,r)}_N of the space of symmetric polynomials in N commuting variables and N anticommuting variables. We prove that the ideal {\mathcal I}^{(k,r)}_N is stable with respect to the action of the negative-half of the super-Virasoro algebra. In addition, we show that the Jack superpolynomials in {\mathcal I}^{(k,r)}_N vanish when k+1 of their commuting variables are equal, and conjecture that they do not vanish when only k of them are identified. This allows us to conclude that the standard Jack polynomials with prescribed symmetry should satisfy similar clustering properties. Finally, we conjecture that the elements of {\mathcal I}^{(k,2)}_N provide a basis for the subspace of symmetric superpolynomials in N variables that vanish when k+1 commuting variables are set equal to each other.

math-ph↗
arXiv2011

Evaluation and normalization of Jack superpolynomials

Two evaluation formulas are derived for the Jack superpolynomials. The evaluation formulas are expressed in terms of products of fillings of skew diagrams. One of these formulas is nothing but the evaluation formula of the Jack polynomials with prescribed symmetry, which thereby receives here a remarkably simple formulation. Among the auxiliary results required to establish the evaluation formulas, the determination of the conditions ensuring the non-vanishing coefficients in a Pieri-type rule for Jack superpolynomials is worth pointing out. An important application of the evaluation formulas is a new derivation of the combinatorial norm of the Jack superpolynomials. We finally mention that the introduction of a simpler version of the dominance ordering on superpartitions is fundamental to establish our results.

math.CO↗
arXiv2010

k-shape poset and branching of k-Schur functions

We give a combinatorial expansion of a Schubert homology class in the affine Grassmannian Gr_{SL_k} into Schubert homology classes in Gr_{SL_{k+1}}. This is achieved by studying the combinatorics of a new class of partitions called k-shapes, which interpolates between k-cores and k+1-cores. We define a symmetric function for each k-shape, and show that they expand positively in terms of dual k-Schur functions. We obtain an explicit combinatorial description of the expansion of an ungraded k-Schur function into k+1-Schur functions. As a corollary, we give a formula for the Schur expansion of an ungraded k-Schur function.

math.CO↗
arXiv2008

Affine insertion and Pieri rules for the affine Grassmannian

We study combinatorial aspects of the Schubert calculus of the affine Grassmannian Gr associated with SL(n,C). Our main results are: 1) Pieri rules for the Schubert bases of H^*(Gr) and H_*(Gr), which expresses the product of a special Schubert class and an arbitrary Schubert class in terms of Schubert classes. 2) A new combinatorial definition for k-Schur functions, which represent the Schubert basis of H_*(Gr). 3) A combinatorial interpretation of the pairing between homology and cohomology of the affine Grassmannian. These results are obtained by interpreting the Schubert bases of Gr combinatorially as generating functions of objects we call strong and weak tableaux, which are respectively defined using the strong and weak orders on the affine symmetric group. We define a bijection called affine insertion, generalizing the Robinson-Schensted Knuth correspondence, which sends certain biwords to pairs of tableaux of the same shape, one strong and one weak. Affine insertion offers a duality between the weak and strong orders which does not seem to have been noticed previously. Our cohomology Pieri rule conjecturally extends to the affine flag manifold, and we give a series of related combinatorial conjectures.

math.CO↗
arXiv2008

A normalization formula for the Jack polynomials in superspace and an identity on partitions

We prove a previously conjectured closed form formula for the norm of the Jack polynomials in superspace with respect to a certain scalar product. The proof is mainly combinatorial and relies on the explicit expression in terms of admissible tableaux of the non-symmetric Jack polynomials. In the final step of the proof appears an identity on weighted sums of partitions that we demonstrate using the methods of Gessel-Viennot.

math.CO↗
arXiv2007

A recursion formula for k-Schur functions

The Bernstein operators allow to build recursively the Schur functions. We present a recursion formula for k-Schur functions at t=1 based on combinatorial operators that generalize the Bernstein operators. The recursion leads immediately to a combinatorial interpretation for the expansion coefficients of k-Schur functions at t=1 in terms of homogeneous symmetric functions.

math.CO↗
arXiv2006

Orthogonality of Jack polynomials in superspace

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 hep-th/0209074 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 math.CO/0509408

math-ph↗
arXiv2005

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.

math.CO↗
arXiv2005

A k-tableau characterization of k-Schur functions

We study k-Schur functions characterized by k-tableaux, proving combinatorial properties such as a k-Pieri rule and a k-conjugation. This new approach relies on developing the theory of k-tableaux, and includes the introduction of a weight-permuting involution on these tableaux that generalizes the Bender-Knuth involution. This work lays the groundwork needed to prove that the set of k-Schur Littlewood-Richardson coefficients contains the 3-point Gromov-Witten invariants; structure constants for the quantum cohomology ring.

math.CO↗
arXiv2004

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.

hep-th↗
arXiv2004

Supersymmetry in the Non-Commutative Plane

The supersymmetric extension of a model introduced by Lukierski, Stichel and Zakrewski in the non-commutative plane is studied. The Noether charges associated to the symmetries are determined. Their Poisson algebra is investigated in the Ostrogradski--Dirac formalism for constrained Hamiltonian systems. It is shown to provide a supersymmetric generalization of the Galilei algebra with a two-dimensional central extension.

hep-th↗
arXiv2004

Order ideals in weak subposets of Young's lattice and associated unimodality conjectures

The k-Young lattice Y^k is a weak subposet of the Young lattice containing partitions whose first part is bounded by an integer k>0. The Y^k poset was introduced in connection with generalized Schur functions and later shown to be isomorphic to the weak order on the quotient of the affine symmetric group by a maximal parabolic subgroup. We prove a number of properties for $Y^k$ including that the covering relation is preserved when elements are translated by rectangular partitions with hook-length $k$. We highlight the order ideal generated by an $m\times n$ rectangular shape. This order ideal, L^k(m,n), reduces to L(m,n) for large k, and we prove it is isomorphic to the induced subposet of L(m,n) whose vertex set is restricted to elements with no more than k-m+1 parts smaller than m. We provide explicit formulas for the number of elements and the rank-generating function of L^k(m,n). We conclude with unimodality conjectures involving q-binomial coefficients and discuss how implications connect to recent work on sieved q-binomial coefficients.

math.CO↗
arXiv2003

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.

hep-th↗
arXiv2003

Generalized Hermite polynomials in superspace as eigenfunctions of the supersymmetric rational CMS model

We present an algebraic construction of the orthogonal eigenfunctions of the supersymmetric extension of the rational Calogero-Moser-Sutherland model with harmonic confinement. These eigenfunctions are the superspace extension of the generalized Hermite (or Hi-Jack) polynomials. The conserved quantities of the rational supersymmetric model are related to their trigonometric relatives through a similarity transformation. This leads to a simple expression between the corresponding eigenfunctions: the generalized Hermite superpolynomials are written as a differential operator acting on the corresponding Jack superpolynomials. As an aside, the maximal superintegrability of the supersymmetric rational Calogero-Moser-Sutherland model is demonstrated.

hep-th↗
arXiv2003

Determinantal Construction of Orthogonal Polynomials Associated with Root Systems

We consider semisimple triangular operators acting in the symmetric component of the group algebra over the weight lattice of a root system. We present a determinantal formula for the eigenbasis of such triangular operators. This determinantal formula gives rise to an explicit construction of the Macdonald polynomials and of the Heckman-Opdam generalized Jacobi polynomials.

math.CO↗
arXiv1996

Rodrigues formulas for the Macdonald polynomials

We present formulas of Rodrigues type giving the Macdonald polynomials for arbitrary partitions through the repeated application of creation operators on the constant 1. Three expressions for the creation operators are derived one from the other. When the last of these expressions is used, the associated Rodrigues formula readily implies the integrality of the (q,t)-Kostka coefficients. The proofs given in this paper rely on the connection between affine Hecke algebras and Macdonald polynomials

q-alg↗