arXiv subjects

L. Lapointe

Publications and source records attributed to L. Lapointe.

At least 19 recordsLinked to original sources

arXiv2018

Bernstein operators and super-Schur functions: combinatorial aspects

The Bernstein vertex operators, which can be used to build recursively the Schur functions, are extended to superspace. Four families of super vertex operators are defined, corresponding to the four natural families of Schur functions in superspace. Combinatorial proofs that the super Bernstein vertex operators indeed build the Schur functions in superspace recursively are provided. We briefly mention a possible realization, in terms of symmetric functions in superspace, of the super-KP hierarchy, where the tau-function naturally expands in one of the super-Schur bases.

math-ph↗

arXiv2017

Pieri rules for the Jack polynomials in superspace and the 6-vertex model

We present Pieri rules for the Jack polynomials in superspace. The coefficients in the Pieri rules are, except for an extra determinant, products of quotients of linear factors in $α$ (expressed, as in the usual Jack polynomial case, in terms of certain hook-lengths in a Ferrers' diagram). We show that, surprisingly, the extra determinant is related to the partition function of the 6-vertex model. We give, as a conjecture, the Pieri rules for the Macdonald polynomials in superspace.

math.CO↗

arXiv2017

N>=2 symmetric superpolynomials

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.

math-ph↗

arXiv2015

Magnetic structure of the antiferromagnetic half-Heusler compound NdBiPt

We present results of single crystal neutron diffraction experiments on the rare-earth, half-Heusler antiferromagnet (AFM) NdBiPt. This compound exhibits an AFM phase transition at $T_{\mathrm N}=2.18$~K with an ordered moment of $1.78(9)$~$μ_{\mathrm B}$ per Nd atom. The magnetic moments are aligned along the $[001]$-direction, arranged in a type-I AFM structure with ferromagnetic planes, alternating antiferromagnetically along a propagation vector $τ$ of $(100)$. The $R$BiPt ($R$= Ce-Lu) family of materials has been proposed as candidates of a new family of antiferromagnetic topological insulators (AFTI) with magnetic space group that corresponds to a type-II AFM structure where ferromagnetic sheets are stacked along the space diagonal. The resolved structure makes it unlikely, that NdBiPt qualifies as an AFTI.

cond-mat.str-el↗

arXiv2014

Magnetic structure of GdBiPt: A candidate antiferromagnetic topological insulator

A topological insulator is a state of matter which does not break any symmetry and is characterized by topological invariants, the integer expectation values of non-local operators. Antiferromagnetism on the other hand is a broken symmetry state in which the translation symmetry is reduced and time reversal symmetry is broken. Can these two phenomena coexist in the same material? A proposal by Mong {\it et al.}\cite{Mong2010} asserts that the answer is yes. Moreover, it is theoretically possible that the onset of antiferromagnetism enables the non-trivial topology since it may create spin-orbit coupling effects which are absent in the non-magnetic phase. The current work examines a real system, half-Heusler GdBiPt, as a candidate for topological anti ferromagnetism. We find that the magnetic moments of the gadolinium atoms form ferromagnetic sheets which are stacked antiferromagnetically along the body diagonal. This magnetic structure may induce spin orbit coupling on band electrons as they hop perpendicular to the ferromagnetic sheets.

cond-mat.str-el↗

arXiv2013

Macdonald polynomials in superspace as eigenfunctions of commuting operators

A generalization of the Macdonald polynomials depending upon both commuting and anticommuting variables has been introduced recently. The construction relies on certain orthogonality and triangularity relations. Although many superpolynomials were constructed as solutions of highly over-determined system, the existence issue was left open. This is resolved here: we demonstrate that the underlying construction has a (unique) solution. The proof uses, as a starting point, the definition of the Macdonald superpolynomials in terms of the Macdonald non-symmetric polynomials via a non-standard (anti)symmetrization and a suitable dressing by anticommuting monomials. This relationship naturally suggests the form of two family of commuting operators that have the defined superpolynomials as their common eigenfunctions. These eigenfunctions are then shown to be triangular and orthogonal. Up to a normalization, these two conditions uniquely characterize these superpolynomials. Moreover, the Macdonald superpolynomials are found to be orthogonal with respect to a second (constant-term-type) scalar product and its norm is evaluated. The latter is shown to match (up to a q-power) the conjectured norm with respect to the original scalar product. Finally, we recall the super-version of the Macdonald positivity conjecture and present two new conjectures which both provide a remarkable relationship between the new (q,t)-Kostka coefficients and the usual ones.

math-ph↗

arXiv2013

Super-Whittaker vector at c=3/2

The degenerate Whittaker vector of the superconformal algebra can be represented in terms of Jack superpolynomials. However, in this representation the norm of the Whittaker vector involves a scalar product with respect to which the Jack superpolynomials are not orthogonal. In this note, we point out that this defect can be cured at c=3/2 by means of a trick specific to the supersymmetric case. At c=3/2, we thus end up with a closed-form expression for the norm of the degenerate super-Whittaker vector. Granting the super-version of the AGT conjecture, this closed-form expression should be equal to the Z_2-symmetric SU(2) pure-gauge instanton partition function -- the corresponding equality taking the form of a rather nontrivial combinatorial identity.

hep-th↗

arXiv2013

Double Macdonald polynomials as the stable limit of Macdonald superpolynomials

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 is sufficiently large: their decomposition in the monomial basis is then independent of 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-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, positivity, etc). When lifted back to superspace, this validates various previously formulated conjectures in the stable regime. The q,t-Kostka coefficients associated to the double Macdonald polynomials are shown to be q,t-analogs of the dimensions of the irreducible representations of the hyperoctahedral group 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, a formula again characteristic of the Coxeter group of type B_n. Finally, as a side result, we obtain a simple identity involving products of four Littlewood-Richardson coefficients.

math-ph↗

arXiv2011

Macdonald polynomials in superspace: conjectural definition and positivity conjectures

We introduce a conjectural construction for an extension to superspace of the Macdonald polynomials. The construction, which depends on certain orthogonality and triangularity relations, is tested for high degrees. We conjecture a simple form for the norm of the Macdonald polynomials in superspace, and a rather non-trivial expression for their evaluation. We study the limiting cases q=0 and q=\infty, which lead to two families of Hall-Littlewood polynomials in superspace. We also find that the Macdonald polynomials in superspace evaluated at q=t=0 or q=t=\infty seem to generalize naturally the Schur functions. In particular, their expansion coefficients in the corresponding Hall-Littlewood bases appear to be polynomials in t with nonnegative integer coefficients. More strikingly, we formulate a generalization of the Macdonald positivity conjecture to superspace: the expansion coefficients of the Macdonald superpolynomials expanded into a modified version of the Schur superpolynomial basis (the q=t=0 family) are polynomials in q and t with nonnegative integer coefficients.

math-ph↗

arXiv2005

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 \hat{su}(\ell) 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 Shimozono's 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.

math.CO↗

arXiv2004

Symmetric functions in superspace

We construct a generalization of the theory of symmetric functions 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 acting on the sets of commuting and anticommuting variables. We first obtain superspace analogues of a number of standard objects and concepts in the theory of symmetric functions: partitions, monomials, elementary symmetric functions, completely symmetric functions, power sums, involutions, generating functions, Cauchy formulas, and scalar products. We then consider a one-parameter extension of the combinatorial scalar product. It provides the natural setting for the definition of a family of ``combinatorial'' orthogonal Jack polynomials in superspace. We show that this family coincides with that of ``physical'' Jack polynomials in superspace that were previously introduced by the authors as orthogonal eigenfunctions of a supersymmetric quantum mechanical many-body problem. The equivalence of the two families is established by showing that the ``physical'' Jack polynomials are also orthogonal with respect to the combinatorial scalar product. This equivalence is also directly demonstrated for particular values of the free parameter.

math.CO↗

arXiv2004

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

The $k$-Young lattice $Y^k$ is a partial order on partitions with no part larger than $k$. This weak subposet of the Young lattice originated from the study of the $k$-Schur functions(atoms) $s_λ^{(k)}$, symmetric functions that form a natural basis of the space spanned by homogeneous functions indexed by $k$-bounded partitions. The chains in the $k$-Young lattice are induced by a Pieri-type rule experimentally satisfied by the $k$-Schur functions. Here, using a natural bijection between $k$-bounded partitions 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 group $\tilde S_{k+1}$ by a maximal parabolic subgroup. From this, the conjectured $k$-Pieri rule implies that the $k$-Kostka matrix connecting the homogeneous basis $\{h_\la\}_{\la\in\CY^k}$ to $\{s_\la^{(k)}\}_{\la\in\CY^k}$ 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.

math.CO↗

arXiv2003

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.

hep-th↗

arXiv2003

Supersymmetric Calogero-Moser-Sutherland models and Jack superpolynomials

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.

hep-th↗

arXiv2003

Supersymmetric Calogero-Moser-Sutherland models: superintegrability structure and eigenfunctions

We first review the construction of the supersymmetric extension of the (quantum) Calogero-Moser-Sutherland (CMS) models. We stress the remarkable fact that this extension is completely captured by the insertion of a fermionic exchange operator in the Hamiltonian: sCMS models ({\it s} for supersymmetric) are nothing but special exchange-type CMS models. Under the appropriate projection, the conserved charges can thus be formulated in terms of the standard Dunkl operators. This is illustrated in the rational case, where the explicit form of the 4N (N being the number of bosonic variables) conserved charges is presented, together with their full algebra. The existence of 2N commuting bosonic charges settles the question of the integrability of the srCMS model. We then prove its superintegrability by displaying 2N-2 extra independent charges commuting with the Hamiltonian. In the second part, we consider the supersymmetric version of the trigonometric case (stCMS model) and review the construction of its eigenfunctions, the Jack superpolynomials. This leads to closed-form expressions, as determinants of determinants involving supermonomial symmetric functions. Here we focus on the main ideas and the generic aspects of the construction: those applicable to all models whether supersymmetric or not. Finally, the possible Lie superalgebraic structure underlying the stCMS model and its eigenfunctions is briefly considered.

hep-th↗

arXiv2002

Jack superpolynomials, superpartition ordering and determinantal formulas

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 model's 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.

hep-th↗

arXiv2002

Fusion bases as facets of polytopes

A new way of constructing fusion bases (i.e., the set of inequalities governing fusion rules) out of fusion elementary couplings is presented. It relies on a polytope reinterpretation of the problem: the elementary couplings are associated to the vertices of the polytope while the inequalities defining the fusion basis are the facets. The symmetry group of the polytope associated to the lowest rank affine Lie algebras is found; it has order 24 for $\su(2)$, 432 for $\su(3)$ and quite surprisingly, it reduces to 36 for $\su(4)$, while it is only of order 4 for $\sp(4)$. This drastic reduction in the order of the symmetry group as the algebra gets more complicated is rooted in the presence of many linear relations between the elementary couplings that break most of the potential symmetries. For $\su(2)$ and $\su(3)$, it is shown that the fusion-basis defining inequalities can be generated from few (1 and 2 respectively) elementary ones. For $\su(3)$, new symmetries of the fusion coefficients are found.

hep-th↗

arXiv2001

Tableau atoms and a new Macdonald positivity conjecture

Let $Λ$ be the space of symmetric functions and $V_k$ be the subspace spanned by the modified Schur functions $\{S_λ[X/(1-t)]\}_{λ_1\leq k}$. We introduce a new family of symmetric polynomials, $\{A_λ^{(k)}[X;t]\}_{λ_1\leq k}$, constructed from sums of tableaux using the charge statistic. We conjecture that the polynomials $A_λ^{(k)}[X;t]$ form a basis for $V_k$ and that the Macdonald polynomials indexed by partitions whose first part is not larger than $k$ expand positively in terms of our polynomials. A proof of this conjecture would not only imply the Macdonald positivity conjecture, but would substantially refine it. Our construction of the $A_λ^{(k)}[X;t]$ relies on the use of tableaux combinatorics and yields various properties and conjectures on the nature of these polynomials. Another important development following from our investigation is that the $A_λ^{(k)}[X;t]$ seem to play the same role for $V_k$ as the Schur functions do for $Λ$. In particular, this has led us to the discovery of many generalizations of properties held by the Schur functions, such as Pieri and Littlewood-Richardson type coefficients.

math.QA↗