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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 $\alpha$ (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., Comment: 28 pages

Generalizations of the (rank- 1 1 ) Bannai–Ito algebra are obtained from a refinement of the grade involution of the Lie superalgebra o s p ( 1 , 2 ) \mathfrak {osp}(1,2) . A hyperoctahedral extension is derived by using a realization of o s p ( 1 , 2 ) \mathfrak {osp}(1,2) in terms of Dunkl operators associated with the Weyl group B 3 B_3 .

The Schur functions in superspace $s_\Lambda$ and $\overline{s}_\Lambda$ are the limits $q=t= 0$ and $q=t=\infty$ respectively of the Macdonald polynomials in superspace. We present the elementary properties of the bases $s_\Lambda$ and $\overline{s}_\Lambda$ (which happen to be essentially dual) such as Pieri rules, dualities, monomial expansions, tableaux generating functions, and Cauchy identities., Les fonctions de Schur dans le superespace $s_\Lambda$ et $\overline{s}_\Lambda$ sont les limites $q=t= 0$ et $q=t=\infty$ respectivement des polynômes de Macdonald dans le superespace. Nous présentons les propriétés élémentaires des bases $s_\Lambda$ et $\overline{s}_\Lambda$ (qui sont essentiellement duales l'une de l'autre) tels que les règles de Pieri, la dualité, le développement en fonctions monomiales, les fonctions génératrices de tableaux et les identités de Cauchy.

We introduce generalizations of Macdonald polynomials indexed by n -tuples of partitions and characterized by certain orthogonality and triangularity relations. We prove that they can be explicitly given as products of ordinary Macdonald polynomials depending on special alphabets. With this factorization in hand, we establish their most basic properties, such as explicit formulas for their norm-squared, evaluation and reproducing kernel. Moreover, we show that the q , t -Kostka coefficients associated to the multi-Macdonald polynomials are positive and correspond to q , t -analogs of the dimensions of the irreducible representations of the wreath product C n ∼ S d , where C n is the cyclic group of order n .

To synthesize the available evidence on playground-related injuries and to determine the prevalence of these injuries in pediatric populations.A rapid systematic review was conducted using PubMed, EMBASE, and the Cochrane Library, as well as the gray literature.The search was limited to studies published between 2012 and 2016 and identified a total of 858 articles, of which 22 met our inclusion criteria: original quantitative studies published in peer-reviewed journals in the past 5 years, concerning unintentional injuries in playgrounds in children aged 0 to 18 years.Information was collected on study and injury characteristics, and the proportion of pediatric injuries related to playground activity was determined. Studies were performed in various countries and most were retrospective cohort studies. The prevalence of playground-related injury ranged from 2% to 34% (median 10%). Studies varied in the types of injuries investigated, including head injuries, genitourinary injuries, ocular and dental trauma, and various types of fractures. Most injuries were low severity.Although playgrounds are a common location where pediatric injuries occur, these injuries are relatively low in frequency and severity.

We show that the ring of symmetric functions in superspace is a cocommutative and self-dual Hopf algebra. We provide formulas for the action of the coproduct and the antipode on various bases of that ring. We introduce the ring sQSym of quasisymmetric functions in superspace and show that it is a Hopf algebra. We give explicitly the product, coproduct and antipode on the basis of monomial quasisymmetric functions in superspace. We prove that the Hopf dual of sQSym, the ring sNSym of noncommutative symmetric functions in superspace, has a multiplicative basis dual to the monomial quasisymmetric functions in superspace.

The goal of this study was to meet a small, remote emergency department's need to reflect on the minimum threshold of services to offer. The study's main objectives were to 1) provide a statistical profile of the emergency services in Fermont, Quebec, 2) assess the staff's and users' perception of the threshold of services offered and 3) propose solutions for improving care and services.This case study was conducted with a participatory approach and a mixed methodology. We compared the results from a questionnaire on the emergency services that was validated during a previous study with the results concerning the other rural emergency services in Quebec as well as with national and provincial recommendations. The questionnaire concerned users' sociodemographic characteristics, the hospital's and the emergency services' descriptors, the services available locally, and the physician and nurse staff. Interviews were also carried out with 33 people (health care professionals, policy-makers and citizens).Fermont's emergency department is smaller than the average rural emergency department in Quebec. They have resources that are in some respects comparable to those of other emergency departments and in line with the recommendations; in other respects, their resources are rather limited. Respondents emphasized how important it is to take into account the environment's specific features when establishing the minimum threshold of services. The proposed solutions would promote collaboration, break down silos within professional practice and focus on training.Fermont's case aside, this exploratory case study highlights how important it is to adopt a pluralistic, participatory and local approach in order to support reflection on the minimum threshold of services in remote emergency departments and to improve their overall performance.Cette étude visait à répondre au besoin d’une petite urgence éloignée d’assoir sa réflexion autour d’un seuil minimum de services à offrir. Les principaux objectifs de l’étude étaient de : 1) dresser le portrait statistique de l’urgence de Fermont, Québec, 2) évaluer la perception du personnel et des usagers en lien avec le seuil de services offert et 3) énoncer des solutions permettant d’améliorer les soins et les services.Cette étude de cas a été réalisée avec une approche participative et une méthodologie mixte. Nous avons comparée les données d’un questionnaire sur l’urgence qui a été validé lors d’une étude précédente aux données des autres urgences rurales québécoises et à des recommandations nationales et provinciales. Le questionnaire portait sur les caractéristiques sociodémographiques des usagers, les descripteurs du centre hospitalier et de l’urgence, les services disponibles localement et les effectifs médicaux et infirmiers. Des entrevues ont aussi été réalisées auprès de 33 personnes (professionnels de la santé, décideurs et citoyens).L’urgence de Fermont est plus petite que la moyenne des urgences rurales québécoises et dotée de ressources à certains égards comparables aux autres urgences et aux recommandations et à d’autres égards limités. Les répondants ont souligné l’importance de tenir compte des particularités du milieu dans l’établissement d’un seuil minimum de services. Les solutions proposées concernent l’importance de favoriser la collaboration, de décloisonner la pratique professionnelle et de miser sur la formation.Au-delà du cas de Fermont, cette étude de cas exploratoire a permis de souligner l’importance d’adopter une approche pluraliste, participative et locale pour soutenir la réflexion autour du seuil minimum de services à offrir dans les urgences éloignées et pour améliorer la performance générale des urgences rurales.

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.

We demonstrate the validity of previously conjectured explicit expressions for the norm and the evaluation of the Macdonald polynomials in superspace. These expressions, which involve the arm-lengths and leg-lengths of the cells in certain Young diagrams, specialize to the well known formulas for the norm and the evaluation of the usual Macdonald polynomials.

Introduction: Rural trauma patients are at increased risk of morbidity and mortality compared to trauma patients treated in urban facilities. Factors contributing to this disparity include differences in resource availability and increased time to definitive treatment for rural patients. Telemedicine can improve the early management of these patients by enabling rural providers to consult with trauma specialists at urban centres. The purpose of this study was to assess the impact of telemedicine utilisation on the diagnosis, clinical management and outcomes of rural trauma patients. Materials and Methods: A rapid review of the literature was performed using the concepts 'trauma', 'rural' and 'telemedicine'. Fifteen electronic databases were searched from inception to 29th June 2018. Manual searches were also conducted in relevant systematic reviews, key journals and bibliographies of included studies. Results: The literature search identified 187 articles, of which 8 articles were included in the review. All 8 studies reported on clinical management, while the impact of telemedicine use on diagnosis and outcomes was reported in 4 and 5 studies, respectively. Study findings suggest that the use of telemedicine may improve patient diagnosis, streamline the process of transferring patients and reduce length of stay. Use of telemedicine had minimal impact on mortality and complications in rural trauma patients. Conclusions: The evidence identified by this rapid review suggests that telemedicine may improve the diagnosis, management and outcomes of rural trauma patients. Further research is required to validate these findings by performing large and well-designed studies in rural areas, ideally as randomised clinical trials.

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., 18 pages

We prove that structure constants related to Hecke algebras at roots of unity are special cases of k k -Littlewood-Richardson coefficients associated to a product of k 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 s u ^ ( ℓ ) \widehat {su}(\ell ) are shown to be k k -Littlewood-Richardson coefficients. From this, Mark Shimozono conjectured that the k k -Schur functions form the Schubert basis for the homology of the loop Grassmannian, whereas k k -Schur coproducts correspond to the integral cohomology of the loop Grassmannian. We introduce dual k k -Schur functions defined on weights of k 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.

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., 21 pages, this supersedes the first part of math.CO/0412306

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., Comment: 6 pages. To appear in the proceedings of the {\it XIII International Colloquium on Integrable Systems and Quantum Groups}, Czech. J . Phys., June 17-19 2004, Doppler Institute, Czech Technical University

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 Sk + 1 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 x 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.

We obtain general identities for the product of two Schur functions in the case where one of the functions is indexed by a rectangular partition, and give their t-analogs using vertex operators. We study subspaces forming a filtration for the symmetric function space that lends itself to generalizing the theory of Schur functions and also provides a convenient environment for studying the Macdonald polynomials. We use our identities to prove that the vertex operators leave such subspaces invariant. We finish by showing that these operators act simply on the k-Schur functions, thus leading to a concept of irreducibility for these functions., Comment: 18 pages

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., 20 pages - minor corrections; to appear in CMP

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 $1.78(9)$~$\mu_{\mathrm B}$ 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 $\tau$ 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., Comment: As resubmitted to PRB, corrected typos and changed symbols in Fig. 3

The Calogero-Sutherland model occurs in a large number of physical contexts, either directly or via its eigenfunctions, the Jack polynomials. The supersymmetric counterpart 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 ${\widehat{\mathfrak {sl}}_2}$ algebra.

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., Comment: 6 pages, 4 figures. Accepted for publication in Phys. Rev. B Rapid Comm

This book is an exposition of the current state of research of affine Schubert calculus and $k$-Schur functions. This text is based on a series of lectures given at a workshop titled "Affine Schubert Calculus" that took place in July 2010 at the Fields Institute in Toronto, Ontario. The story of this research is told in three parts: 1. Primer on $k$-Schur Functions 2. Stanley symmetric functions and Peterson algebras 3. Affine Schubert calculus

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, 15 pages, AmsLaTeX

Thermal impedance spectroscopy (TIS) is applied for the thermal characterization of a large prismatic battery cell. The method is based on previously published work and is extended to a three dimensional system. By using a discretized three dimensional model, the experimental results can be fitted to obtain the specific heat capacity and heat conductivity of the cell in three dimensions. The results show good agreement with the available data in the literature. Suggestions are made to improve the test setup to gain more confidence in the results and to increase the precision of the model.

We uncover a deep connection between the $\mathcal{N}=1$ superconformal field theory in 2D and eigenfunctions of the supersymmetric Sutherland model known as Jack superpolynomials (sJacks). Specifically, the singular vector at level $rs/2$ of the Kac module labeled by the two integers $r$ and $s$ are given explicitly as a sum of sJacks whose indexing diagrams are contained in a rectangle with $r$ columns and $s$ rows As a second compelling evidence for the distinguished status of the sJack-basis in SCFT, we find that the degenerate Whittaker vectors (Gaiotto states) can be expressed as a remarkably simple linear combination of sJacks. As a consequence, we are able to reformulate the supersymmetric version of the (degenerate) AGT conjecture in terms of the combinatorics of sJacks. Note that the closed-form formulas for the singular vectors and the degenerate Whittaker vectors, although only conjectured in general, have been heavily tested (in some cases, up to level 33/2). Note also that both the Neveu-Schwarz and Ramond sectors are treated., 37 pages. v3: correction of some equations (not corrected in the published version), most importantly, (6.9), (6.10), and (B.27)

The Jack polynomials P_\lambda^{(\alpha)} at \alpha=-(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_{\Lambda}^{(\alpha)} at \alpha=-(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., Comment: 36 pages; the main changes in v2 are : 1) in the introduction, we present exceptions to an often made statement concerning the clustering property of the ordinary Jack polynomials for (k,r,N)-admissible partitions (see Footnote 2); 2) Conjecture 14 is substantiated with the extensive computational evidence presented in the new appendix C; 3) the various tests supporting Conjecture 16 are reported

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., Comment: 40 pages; v2: title and abstract changed; minor modifications in the text

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., Comment: v2: revised and enlarged version with new section on the existence of schur functions in superspace

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., Comment: 42 pages, 10 figures. v2: minor corrections in Eqs (16) and (17)

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., Comment: 20 pages, 7 figures

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., Comment: 98 pages

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., 19 pages

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., Comment: 13 pages, no figures, the version for publication in Ann. Phys

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) $s_\lambda^{(k)}$, 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., Comment: 30 pages, 1 figure

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., 28 pages

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., Latex 2e, shortened version, one reference added, 18 pages

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., 28 pages. Corrected version of lemme 3 and other minor corrections and 2 new references; version to appear in Commun. Math. Phys

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., Comment: Minor corrections in tables; 30 pages

Let $\Lambda$ be the space of symmetric functions and $V_k$ be the subspace spanned by the modified Schur functions $\{S_\lambda[X/(1-t)]\}_{\lambda_1\leq k}$. We introduce a new family of symmetric polynomials, $\{A_{\lambda}^{(k)}[X;t]\}_{\lambda_1\leq k}$, constructed from sums of tableaux using the charge statistic. We conjecture that the polynomials $A_{\lambda}^{(k)}[X;t]$ 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 $A_\lambda^{(k)}[X;t]$ 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 $A_{\lambda}^{(k)}[X;t]$ seem to play the same role for $V_k$ as the Schur functions do for $\Lambda$. 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., Comment: 38 pages, 7 figures. New version, with minor modifications, of "A filtration of the symmetric function space and a refinement of the Macdonald positivity conjecture"

We describe matrices whose determinants are the Jack polynomials expanded in terms of the monomial basis. The top row of such a matrix is a list of monomial functions, the entries of the sub-diagonal are of the form $-(r\alpha+s)$, with $r$ and $s \in {\bf N^+}$, the entries above the sub-diagonal are non-negative integers, and below all entries are 0. The quasi-triangular nature of these matrices gives a recursion for the Jack polynomials allowing for efficient computation. A specialization of these results yields a determinantal formula for the Schur functions and a recursion for the Kostka numbers.

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., 9 pages

The Macdonald polynomials can be obtained by acting on the constant 1 with creation operators. Three different expressions for these operators are derived, one from the other, in a rather succint way. When the last of these expressions is used, the formalism is seen to imply straightforwardly the integrality of the (q,t)-Kostka coefficients, that is of the expansion coefficients for the Macdonald functions in terms of Schur functions., 8 pages, AmsLaTeX

The wave functions of the Calogero-Sutherland model are known to be expressible in terms of Jack polynomials. A formula which allows to obtain the wave functions of the excited states by acting with a string of creation operators on the wave function of the ground state is presented and derived. The creation operators that enter in this formula of Rodrigues-type for the Jack polynomials involve Dunkl operators., 35 pages, LaTeX2e with amslatex

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, Comment: 22 pages, this supersedes the second part of math.CO/0412306; (v2) 24 pages, title and abstract slightly modified, minor changes, typos corrected

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., 18 pages, 3 figures

We consider a filtration of the symmetric function space given by $\Lambda^{(k)}_t$, the linear span of Hall-Littlewood polynomials indexed by partitions whose first part is not larger than $k$. We introduce symmetric functions called the $k$-Schur functions, providing an analog for the Schur functions in the subspaces $\Lambda^{(k)}_t$. We prove several properties for the $k$-Schur functions including that they form a basis for these subspaces that reduces to the Schur basis when $k$ is large. We also show that the connection coefficients for the $k$-Schur function basis with the Macdonald polynomials belonging to $\Lambda^{(k)}_t$ are polynomials in $q$ and $t$ with integral coefficients. In fact, we conjecture that these integral coefficients are actually positive, and give several other conjectures generalizing Schur function theory., Comment: 24 pages