Variétés Kähleriennes Compactes (book) (1970, 90 pp.)
Alain Lascoux, Marcel Berger
@book{lascoux1970varieteskahleriennes,
author = {Alain Lascoux and Marcel Berger},
title = {{Variétés Kähleriennes Compactes (book)}},
year = {1970},
volume = {154},
publisher = {Springer-Verlag},
series = {Lecture Notes in Mathematics},
doi = {10.1007/bfb0069331}
}
Polynômes symétriques et coefficients d'intersection de cycles de Schubert (1974, 4 pp.)
Alain Lascoux
On explicite les liens entre les cycles de Schubert et les polynômes symétrique, en introduisant des opérateurs de translation qui permettent aussi d'étudier la structure des λ-anneaux.
@article{lascoux1974polynomessymetriques,
author = {Alain Lascoux},
title = {{Polynômes symétriques et coefficients d'intersection de cycles de Schubert}},
year = {1974},
journal = {C. R. Acad. Sci. Paris Sér. A-B},
volume = {279},
pages = {201--204},
note = {Présentée par Henri Cartan}
}
Puissances extérieures, déterminants et cycles de Schubert (1974, 19 pp.)
Alain Lascoux
Cet article se propose de montrer qu'étant donné un drapeau de modules localement libres A1 <= ... <= Ap et un module localement libre E sur une base lisse S quasi projective, on peut leur associer des cycles possédant les mêmes propriétés que dans le cas classique ou les Ai sont des modules triviaux, S une grassmannienne sur un corps algébriquement clos, et E le fibré « tautologique » sur la grassmannienne.
@article{lascoux1974puissancesexterieures,
author = {Alain Lascoux},
title = {{Puissances extérieures, déterminants et cycles de Schubert}},
year = {1974},
journal = {Bull. Soc. Math. France},
volume = {102},
pages = {161--179},
doi = {10.24033/bsmf.1776}
}
Tableaux de Young et fonctions de Schur-Littlewood (1974, 7 pp.)
Alain Lascoux
@article{lascoux1974tableauxyoung,
author = {Alain Lascoux},
title = {{Tableaux de Young et fonctions de Schur-Littlewood}},
year = {1974},
journal = {Sém. Delange-Pisot-Poitou, Théorie des nombres},
volume = {16},
note = {Exp. No. A2}
}
Fonctions de Schur et grassmanniennes (1975, 7 pp.)
Alain Lascoux
Nous illustrons par quelques formules les liens étroits existant entre la structure des grassmanniennes et les polynômes symétriques dits de Schur.
@article{lascoux1975fonctionsschur,
author = {Alain Lascoux},
title = {{Fonctions de Schur et grassmanniennes}},
year = {1975},
journal = {C. R. Acad. Sci. Paris Sér. A-B},
volume = {281},
pages = {813--819},
note = {Présentée par Henri Cartan}
}
Calcul de Schur et extensions grassmanniennes des λ-anneaux (1977, 35 pp.)
Alain Lascoux
@incollection{lascoux1977calculschur,
author = {Alain Lascoux},
title = {{Calcul de Schur et extensions grassmanniennes des λ-anneaux}},
year = {1977},
booktitle = {Combinatoire et représentation du groupe symétrique (Strasbourg, 1976)},
volume = {579},
pages = {182--216},
publisher = {Springer, Berlin},
series = {Lecture Notes in Math.},
doi = {10.1007/bfb0090018}
}
Sur une conjecture de H. O. Foulkes (1978, 2 pp.)
Alain Lascoux, Marcel-Paul Schützenberger
One sketches a proof of Foulkes’ conjecture on the polynomials defining Littlewood Q-functions in terms of Schur functions.
@article{lascoux1978conjectureh,
author = {Alain Lascoux and Marcel-Paul Schützenberger},
title = {{Sur une conjecture de H. O. Foulkes}},
year = {1978},
journal = {C. R. Acad. Sci. Paris Sér. A-B},
volume = {286},
pages = {323--324}
}
Syzygies des variétés déterminantales (1978, 36 pp.)
Alain Lascoux
@article{lascoux1978syzygiesvarietes,
author = {Alain Lascoux},
title = {{Syzygies des variétés déterminantales}},
year = {1978},
journal = {Adv. in Math.},
volume = {30},
pages = {202--237},
doi = {10.1016/0001-8708(78)90037-3}
}
Croissance des polynômes de Foulkes-Green (1979, 4 pp.)
Alain Lascoux, Marcel-Paul Schützenberger
@article{lascoux1979croissancepolynomes,
author = {Alain Lascoux and Marcel-Paul Schützenberger},
title = {{Croissance des polynômes de Foulkes-Green}},
year = {1979},
journal = {C. R. Acad. Sci. Paris Sér. A-B},
volume = {288}
}
A New Statistics on Words (1980, 5 pp.)
Alain Lascoux, Marcel-Paul Schützenberger
The study of various statistics on words is a chapter of combinatorics and it has been recently surveyed by Foata [1]. One ot these statistics is the so-called “Major Index” of a word. It plays a role in the theory of representations of the symmetric group. We present here a generalization, the “load”, which we used to prove Foulkes’ conjecture, i.e. to supply a q-analogue of Kostka numbers [2].
@article{lascoux1980newstatistics,
author = {Alain Lascoux and Marcel-Paul Schützenberger},
title = {{A New Statistics on Words}},
year = {1980},
journal = {Ann. Discrete Math.},
volume = {6},
pages = {251--255},
doi = {10.1016/s0167-5060(08)70709-x}
}
Produit de Kronecker des représentations du groupe symétrique (1980, 11 pp.)
Alain Lascoux
@incollection{lascoux1980produitkronecker,
author = {Alain Lascoux},
title = {{Produit de Kronecker des représentations du groupe symétrique}},
year = {1980},
booktitle = {Séminaire d'Algèbre Paul Dubreil et Marie-Paule Malliavin},
volume = {795},
pages = {319--329},
publisher = {Springer, Berlin},
series = {Lecture Notes in Math.},
doi = {10.1007/BFb0090125}
}
Le monoïde plaxique (1981, 29 pp.)
Alain Lascoux, Marcel-Paul Schützenberger
The combinatorial description of representations of the symmetric and linear groups is based on Young tableaux. A decisive advance was the introduction of a multiplication operation on the set of tableaux. We develop this perspective and present the main properties of the "plactic" monoid, which has diverse applications (representations of finite linear groups, Hecke algebras, cohomology of flag varieties).
@incollection{lascoux1981monoideplaxique,
author = {Alain Lascoux and Marcel-Paul Schützenberger},
title = {{Le monoïde plaxique}},
year = {1981},
booktitle = {Non-commutative structures in algebra and geometric combinatorics (Naples, 1978)},
number = {109},
publisher = {CNR, Rome},
series = {Quaderni de "La Ricerca Scientifica"}
}
Polynômes de Kazhdan et Lusztig (1981, 19 pp.)
Alain Lascoux, Marcel-Paul Schützenberger
@incollection{lascoux1981polynomeskazhdan,
author = {Alain Lascoux and Marcel-Paul Schützenberger},
title = {{Polynômes de Kazhdan et Lusztig}},
year = {1981},
booktitle = {Young tableaux and Schur functors in algebra and geometry (Toruń, 1980)},
volume = {87-88},
publisher = {Soc. Math. France},
series = {Astérisque}
}
Classes of Determinantal Varieties Associated with Symmetric and Skew-Symmetric Matrices (1982, 12 pp.)
Tomasz Józefiak, Alain Lascoux, Piotr Pragacz
In this paper the authors compute the classes of subschemes of degenerations of a homomorphism of two fibrations in the Chow ring of the base. -- Translated by M. Ackerman
@article{jozefiak1982classesdeterminantal,
author = {Tomasz Józefiak and Alain Lascoux and Piotr Pragacz},
title = {{Classes of Determinantal Varieties Associated with Symmetric and Skew-Symmetric Matrices}},
year = {1982},
journal = {Math. USSR Izvestija},
volume = {18},
pages = {575--586},
note = {Translated from Izv. Akad. Nauk SSSR Ser. Mat. 45 (1981), no. 9, 662--673},
doi = {10.1070/IM1982v018n03ABEH001400}
}
Polynômes de Schubert (1982, 4 pp.)
Alain Lascoux, Marcel-Paul Schützenberger
We define a family of polynomials generalizing Schur symmetric functions. We examine a subfamily and give explicit relationships in the cohomology ring of the flag manifold.
@article{lascoux1982polynomesschubert,
author = {Alain Lascoux and Marcel-Paul Schützenberger},
title = {{Polynômes de Schubert}},
year = {1982},
journal = {C. R. Acad. Sci. Paris Sér. I Math.},
volume = {294}
}
Structure de Hopf de l'anneau de cohomologie et de l'anneau de Grothendieck d'une variété de drapeaux (1982, 5 pp.)
Alain Lascoux, Marcel-Paul Schützenberger
@article{lascoux1982structurehopf,
author = {Alain Lascoux and Marcel-Paul Schützenberger},
title = {{Structure de Hopf de l'anneau de cohomologie et de l'anneau de Grothendieck d'une variété de drapeaux}},
year = {1982},
journal = {C. R. Acad. Sci. Paris Sér. I Math.},
volume = {295}
}
Symmetry and Flag Manifolds (1983, 27 pp.)
Alain Lascoux, Marcel-Paul Schützenberger
@incollection{lascoux1983symmetryflag,
author = {Alain Lascoux and Marcel-Paul Schützenberger},
title = {{Symmetry and Flag Manifolds}},
year = {1983},
booktitle = {Invariant theory (Montecatini, 1982)},
volume = {996},
publisher = {Springer},
series = {Lecture Notes in Math.},
doi = {10.1007/bfb0063238}
}
Caractéristique d'Euler-Poincaré et produit des caractères (1984, 4 pp.)
Alain Lascoux
We show that a formula of Stanley concerning the inner product of Schur functions (i.e. the product of characters of the symmetric group) is linked with an extension of the Euler-Poincaré characteristics of modules on a Grassmann variety.
@article{lascoux1984caracteristiqued,
author = {Alain Lascoux},
title = {{Caractéristique d'Euler-Poincaré et produit des caractères}},
year = {1984},
journal = {C. R. Acad. Sci. Paris Sér. I Math.},
volume = {299},
number = {10},
pages = {447--450},
note = {Présentée par Pierre Deligne}
}
Équerres et fonctions de Schur (1984, 4 pp.)
Alain Lascoux, Piotr Pragacz
We express a skew Schur function as a determinant of hooks.
@article{lascoux1984equerresfonctions,
author = {Alain Lascoux and Piotr Pragacz},
title = {{Équerres et fonctions de Schur}},
year = {1984},
journal = {C. R. Acad. Sci. Paris Sér. I Math.}
}
Interpolation de Newton à plusieurs variables (1985, 15 pp.)
Alain Lascoux, Marcel-Paul Schützenberger
@incollection{lascoux1985interpolationnewton,
author = {Alain Lascoux and Marcel-Paul Schützenberger},
title = {{Interpolation de Newton à plusieurs variables}},
year = {1985},
booktitle = {Séminaire d'algèbre Paul Dubreil et Marie-Paule Malliavin},
volume = {1146},
publisher = {Springer},
series = {Lecture Notes in Math.},
doi = {10.1007/bfb0074538}
}
Schubert Polynomials and the Littlewood—Richardson Rule (1985, 14 pp.)
Alain Lascoux, Marcel-Paul Schützenberger
The decomposition of a product of two irreducible representations of a linear group Gl(N, C) is explicitly given by the Littlewood—Richardson rule, which amounts to finding how many Young tableaux satisfy certain conditions. We obtain more general multiplicities by generating ‘vexillary’ permutations and by using partially symmetrical polynomials (Schubert polynomials).
@article{lascoux1985schubertpolynomials,
author = {Alain Lascoux and Marcel-Paul Schützenberger},
title = {{Schubert Polynomials and the Littlewood—Richardson Rule}},
year = {1985},
journal = {Lett. Math. Phys.},
volume = {10},
pages = {111--124},
doi = {10.1007/bf00398147}
}
La résultante de deux polynômes (1986, 17 pp.)
Alain Lascoux
Aprés avoir rétabli l'équation résultante de deux polynômes dans son sexe originel, nous montrons que la théorie des fonctions symétriques fournit sans calcul de nombreuses expressions déterminantales de la dite résultante, et plus généralement, du plus grand commun diviseur.
La résultante de deux polynômes a été l'objet de maintes recherches depuis le 18éme siècle, étant au coeur de la théorie de l'élimination. On la trouvait développée dans tous les livres d'algèbre du siècle dernier. De nos jours, sa part s'est malheureusement réduite à une expression déterminantale attribuée à Sylvester. Voilà sans doute qui explique la redécouverte périodique de certaines de ses propriétés [cf. Gerber, Householder]. Nous puiserons la majorité de nos références dans les cinq livres de [Muir] qui résument la majorité des formules déterminantales de la littérature (mathématique) jusqu'en 1920.
@incollection{lascoux1986resultantedeux,
author = {Alain Lascoux},
title = {{La résultante de deux polynômes}},
year = {1986},
booktitle = {Séminaire d'Algèbre Paul Dubreil et Marie-Paule Malliavin},
volume = {1220},
pages = {52--72},
publisher = {Springer, Berlin},
series = {Lecture Notes in Math.},
doi = {10.1007/BFb0099505}
}
Suites récurrentes linéaires (1986, 8 pp.)
Alain Lascoux
@article{lascoux1986suitesrecurrentes,
author = {Alain Lascoux},
title = {{Suites récurrentes linéaires}},
year = {1986},
journal = {Adv. in Appl. Math.},
volume = {7},
number = {2},
pages = {228--235},
doi = {10.1016/0196-8858(86)90034-5}
}
Forme canonique d'une forme binaire (1987, 8 pp.)
Alain Lascoux
The space of polynomials in one variable x of odd degree 2n+l is 2n+2-dimensional. A linear combination of n+l polynomials (x-~)2n+l takescare of the required number of parameters. Such an expression has been called the canon- ical form of a binary form. Sylvester has shown that this reduction is obtained through another polynomial, the catalecticant~when it has the taste of having all its roots distinct~ It happens, and this is not by coincidence, that the catalectic- ant is an orthogonal polynomial for a certain linear functional associated to the polynomial ; the required relations are given by the theory of symmetric functions, supplemented by a little amount of divided differences and Lagrange interpolation formula. Having recovered with these tools the classical case, one can go further and treat as well the case where all the roots of the eatalecticant are the same; this is what we do in Theorem 3.4
Symmetrization Operators on Polynomial Rings (1987, 3 pp.)
Alain Lascoux, Marcel-Paul Schützenberger
Any permutation ζ of an ordered alphabet A = {a1, ..., an+1} determines an operator f → f ζ of the polynomial ring C[A] such that c → c, c ∈ C, ai → aζ(i) for i = 1, ..., n + 1. The note deals with the operators D: C[A] → C[A] which can be written in the form D = Σζ ζRζ, the coefficients Rζ being rational functions. The authors announce the following results: (1) a characterisation of operators of the ring C[a1, a2, a3] satisfying some additional conditions; (2) a description of the C[A]-module E of the operators D; (3) a Leibniz-type formula. (Review by Arkadiusz Płoski, Mathematical Reviews MR925080)
@article{lascoux1987symmetrizationoperators,
author = {Alain Lascoux and Marcel-Paul Schützenberger},
title = {{Symmetrization Operators on Polynomial Rings}},
year = {1987},
journal = {Funct. Anal. Appl.},
volume = {21},
number = {4},
pages = {324--326},
doi = {10.1007/bf01077811}
}
Arêtes et tableaux (1988, 10 pp.)
Alain Lascoux, Marcel-Paul Schützenberger
@article{lascoux1988aretestableaux,
author = {Alain Lascoux and Marcel-Paul Schützenberger},
title = {{Arêtes et tableaux}},
year = {1988},
journal = {Publ. I.R.M.A. Strasbourg},
number = {372/S-20},
pages = {109--120},
note = {Actes du 20e Séminaire Lotharingien de Combinatoire (Alghero, 1988)}
}
Ribbon Schur Functions (1988, 17 pp.)
Alain Lascoux, Piotr Pragacz
We present a new determinantal expression for Schur functions. Previous expressions were due to Jacobi, Trudi, Giambelli and others (see (7)) and involved elementary symmetric functions or hook functions. We give, in Theorem 1.1, a decomposition of a Schur function into ribbon functions (also called skew hook functions, new functions by MacMahon, and MacMahon functions by others). We provide two different proofs of this result in Sections 2 and 3. In Section 2, we use Bazin's formula for the minors of a general matrix, as we already did in [6], to decompose a skew Schur function into hooks. In Section 3, we show how to pass from hooks to ribbons and conversely. In Section 4, we generalize to skew Schur functions. In Section 5, we give some applications, and show how such constructions, in the case of staircase partitions, generalize the classical continued fraction for the tangent function due to Euler.
@article{lascoux1988ribbonschur,
author = {Alain Lascoux and Piotr Pragacz},
title = {{Ribbon Schur Functions}},
year = {1988},
journal = {European J. Combin.},
volume = {9},
pages = {561--574},
doi = {10.1016/s0195-6698(88)80053-2}
}
S-Function Series (1988, 10 pp.)
Alain Lascoux, Piotr Pragacz
The authors describe the different methods of generating S-function series, giving some new series as an example in each case.
@article{lascoux1988sfunction,
author = {Alain Lascoux and Piotr Pragacz},
title = {{S-Function Series}},
year = {1988},
journal = {J. Phys. A},
volume = {21},
number = {22},
pages = {4105--4114},
doi = {10.1088/0305-4470/21/22/014}
}
Fonctorialité des polynômes de Schubert (1989, 14 pp.)
Alain Lascoux, Marcel-Paul Schützenberger
The simultaneous lifting of the classical (i.e. commutative) Schubert polynomials into two non commutative algebras related by the "Dualité de Cauchy" gives a non commutative, as well as functorial, definition of these polynomials.
@incollection{lascoux1989fonctorialitepolynomes,
author = {Alain Lascoux and Marcel-Paul Schützenberger},
title = {{Fonctorialité des polynômes de Schubert}},
year = {1989},
booktitle = {Invariant Theory},
volume = {88},
pages = {585--598},
publisher = {Amer. Math. Soc., Providence, RI},
series = {Contemp. Math.},
doi = {10.1090/conm/088/1000001}
}
On Giambelli's theorem on complete correlations (1989, 57 pp.)
Dan Laksov, Alain Lascoux, Anders Thorup
@article{laksov1989giambellis,
author = {Dan Laksov and Alain Lascoux and Anders Thorup},
title = {{On Giambelli's theorem on complete correlations}},
year = {1989},
journal = {Acta Math.},
volume = {162},
pages = {143--199},
doi = {10.1007/BF02392836}
}
Tableaux and Noncommutative Schubert Polynomials (1989, 3 pp.)
Alain Lascoux, Marcel-Paul Schützenberger
@article{lascoux1989tableauxnoncommutative,
author = {Alain Lascoux and Marcel-Paul Schützenberger},
title = {{Tableaux and Noncommutative Schubert Polynomials}},
year = {1989},
journal = {Funktsional. Anal. i Prilozhen.},
volume = {23},
number = {3},
pages = {63--64},
note = {English translation in Funct. Anal. Appl. 23 (1989), no. 3, 223--225},
doi = {10.1007/bf01079531}
}
Anneau de Grothendieck de la variété de drapeaux (1990, 34 pp.)
Alain Lascoux
@incollection{lascoux1990anneaugrothendieck,
author = {Alain Lascoux},
title = {{Anneau de Grothendieck de la variété de drapeaux}},
year = {1990},
booktitle = {The Grothendieck Festschrift, Vol. III},
volume = {88},
pages = {1--34},
publisher = {Birkhäuser},
series = {Progress in Mathematics},
doi = {10.1007/978-0-8176-4576-2_1}
}
Inversion des matrices de Hankel (1990, 26 pp.)
Alain Lascoux
@article{lascoux1990inversionmatrices,
author = {Alain Lascoux},
title = {{Inversion des matrices de Hankel}},
year = {1990},
journal = {Linear Algebra Appl.},
volume = {129},
note = {Submitted by Richard A. Brualdi},
doi = {10.1016/0024-3795(90)90299-r}
}
Keys and standard bases (1990, 21 pp.)
Alain Lascoux, Marcel-Paul Schützenberger
@incollection{lascoux1990keysstandard,
author = {Alain Lascoux and Marcel-Paul Schützenberger},
title = {{Keys and standard bases}},
year = {1990},
booktitle = {Invariant Theory and Tableaux (Minneapolis, MN, 1988)},
volume = {19},
pages = {125--144},
publisher = {Springer, New York},
series = {IMA Vol. Math. Appl.}
}
Wronski’s Factorization of Polynomials (1990, 8 pp.)
Alain Lascoux
In the middle of Wronski‘s papers [8], [9], [10], [11] one can come across solutions of important problems such as finding, for any polynomial with complex coefficients, the factor which corresponds to roots of modulus less than 1.
@incollection{lascoux1990wronskis,
author = {Alain Lascoux},
title = {{Wronski’s Factorization of Polynomials}},
year = {1990},
booktitle = {Topics in Algebra, Part 2 (Warsaw, 1988)},
volume = {26},
publisher = {PWN, Warsaw},
series = {Banach Center Publ.},
doi = {10.4064/-26-2-379-386}
}
Cyclic Permutations on Words, Tableaux and Harmonic Polynomials (1991, 23 pp.)
Alain Lascoux
The compatibility of tableaux with cyclic permutations provides a rank poset structure on the set of tableaux with a given evaluation, allowing the embedding of different sets of tableaux into each other. There results a decomposition of these sets into smaller sets, the cardinals of which are multiplicities of isotypic components (for the symmetric group action) either in cohomology rings of flag manifolds or equivalently in spaces of harmonic polynomials.
@incollection{lascoux1991cyclicpermutations,
author = {Alain Lascoux},
title = {{Cyclic Permutations on Words, Tableaux and Harmonic Polynomials}},
year = {1991},
booktitle = {Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989)},
pages = {323--347},
publisher = {Manoj Prakashan},
address = {Madras}
}
Opérateurs différentiels sur l'anneau des polynômes symétriques (1991, 30 pp.)
Alain Lascoux
@unpublished{lascoux1991operateursdifferentiels,
author = {Alain Lascoux},
title = {{Opérateurs différentiels sur l'anneau des polynômes symétriques}},
year = {1991},
note = {Unpublished}
}
Décompositions dans l’algébre des differences divisées (1992, 15 pp.)
Alain Lascoux, Marcel-Paul Schützenberger
Lascoux, A. et M.-P. Schiitzenberger, Décompositions dans l’algebre des diiférences divisées, Discrete Mathematics 99 (1992) 165—179. The group algebra of the symmetric group on the ring of rational functions has, apart from its canonical basis of permutations, several bases of symmetrizing operators, among which the classical Newton’s divided diflerences. We deal here with the explicitation of the matrices of change of bases. Our main result is that, in the case of Gl(n), the components of the matrices associated to the classical bases are just specializations of Schubert or Grothendieck polynomials. We refer to the work of Arabia, Bernstein—Gelfand—Gelfand, Kac, Kostant, Kumar, Rossmann for the interpretation of these matrices in terms of (equivariant) cohomology and K-theory rings of the flag manifold, or equivariant singularities of Schubert varieties. Our main technical tool is the simple observation that all but one specialization of the maximal twofold Schubert polynomial 2 vanish.
@article{lascoux1992decompositionsdans,
author = {Alain Lascoux and Marcel-Paul Schützenberger},
title = {{Décompositions dans l’algébre des differences divisées}},
year = {1992},
journal = {Discrete Math.},
volume = {99},
number = {1-3},
pages = {165--179},
doi = {10.1016/0012-365x(92)90372-m}
}
Turbo-Straightening for Decomposition into Standard Bases (1992, 17 pp.)
Christophe Carre, Alain Lascoux, Bernard Leclerc
Specht and Hodge have shown that the space generated by products of minors of a matrix admits a linear basis in bijection with Young tableaux. The decomposition of any element into this basis is called straightening and corresponds to the iterative use of Plücker relations. Thanks to a well known isomorphism between the space of harmonic polynomials and the space of polynomials modulo the ideal generated by symmetric polynomials, we can now use as a main technical tool the canonical scalar product on this later space. This leads to a different, and possibly better, algorithm for straightening.
@article{carre1992turbostraightening,
author = {Christophe Carre and Alain Lascoux and Bernard Leclerc},
title = {{Turbo-Straightening for Decomposition into Standard Bases}},
year = {1992},
journal = {Internat. J. Algebra Comput.},
volume = {2},
number = {3},
pages = {275--290},
doi = {10.1142/S0218196792000165}
}
A Pieri Formula in the Grothendieck Ring of a Flag Bundle (1994, 19 pp.)
William Fulton, Alain Lascoux
Green Polynomials and Hall-Littlewood Functions at Roots of Unity (1994, 12 pp.)
Alain Lascoux, Bernard Leclerc, Jean-Yves Thibon
— We give some applications of our recent work [LLT] about Hall-Littlewood functions at roots of unity. In particular, we prove the two conjectures of N. Sultana [Su] on specializations of Green polynomials, and we generalize results of Kraskiewicz-Weyman and Molev-Tsalenko concerning characters of the symmetric group induced by maximal cyclic subgroups. We also give some new congruences for Kostka-Foulkes polynomials modulo cyclotomic polynomials.
@article{lascoux1994greenpolynomials,
author = {Alain Lascoux and Bernard Leclerc and Jean-Yves Thibon},
title = {{Green Polynomials and Hall-Littlewood Functions at Roots of Unity}},
year = {1994},
journal = {European J. Combin.},
volume = {15},
pages = {173--180},
doi = {10.1006/eujc.1994.1019}
}
Crystal Graphs and q-Analogues of Weight Multiplicities for the Root System A_n (1995, 16 pp.)
Alain Lascoux, Bernard Leclerc, Jean-Yves Thibon
We give an expression of the q-analogues of the multiplicities of weights in ir- reducible sl n+1 -modules in terms of the geometry of the crystal graph attached to the corresponding Uq (sl n+1 )-modules. As an application, we describe multivari- ate polynomial analogues of the multiplicities of the zero weight, refining Kostant’s generalized exponents.
@article{lascoux1995crystalgraphs,
author = {Alain Lascoux and Bernard Leclerc and Jean-Yves Thibon},
title = {{Crystal Graphs and q-Analogues of Weight Multiplicities for the Root System A_n}},
year = {1995},
journal = {Lett. Math. Phys.},
volume = {35},
pages = {359--374},
doi = {10.1007/BF00750843}
}
Euler-Poincare Characteristic and Polynomial Representations of Iwahori-Hecke Algebras (1995, 25 pp.)
Gerard Duchamp, Daniel Krob, Alain Lascoux, Bernard Leclerc, Thomas Scharf, Jean-Yves Thibon
The Hecke algebras of type A „ admit faithful representations by symmetrization operators acting on polynomial rings. These operators are related to the geometry of flag manifolds and in particular to a generalized Euler-Poincare characteristic denned by Hirzebruch. They provide g-idempotents, togetherwith a simple way to describe the irreducible representations of the Hecke algebra. The link with Kazhdan-Lusztig representations is discussed. We specially detail the case of hook representations, and as an application, we investigate the hamiltonian of a quantum spin chain with C/ g (su(l/l)) symmetry.
@article{duchamp1995eulerpoincare,
author = {Gerard Duchamp and Daniel Krob and Alain Lascoux and Bernard Leclerc and Thomas Scharf and Jean-Yves Thibon},
title = {{Euler-Poincare Characteristic and Polynomial Representations of Iwahori-Hecke Algebras}},
year = {1995},
journal = {Publ. Res. Inst. Math. Sci.},
volume = {31},
pages = {179--201},
doi = {10.2977/prims/1195164438}
}
Noncommutative Symmetric Functions (1995, 111 pp.)
Israel M. Gelfand, Daniel Krob, Alain Lascoux, Bernard Leclerc, Vladimir Retakh, Jean-Yves Thibon
@article{gelfand1995noncommutativesymmetric,
author = {Israel M. Gelfand and Daniel Krob and Alain Lascoux and Bernard Leclerc and Vladimir Retakh and Jean-Yves Thibon},
title = {{Noncommutative Symmetric Functions}},
year = {1995},
journal = {Adv. Math.},
volume = {112},
pages = {218--348},
doi = {10.1006/aima.1995.1032}
}
Polynômes de Kazhdan-Lusztig pour les variétés de Schubert vexillaires (1995, 5 pp.)
Alain Lascoux
Nous donnons, dans le cas du groupe symétrique, les polynômes de Kazhdan-Lusztig P_μ(ν) pour toutes les permutations vexillaires μ et pour ν quelconque. Nous utilisons pour ce faire la description de l'ordre de Bruhat à l'aide du treillis enveloppant du groupe symétrique.
@article{lascoux1995polynomeskazhdan,
author = {Alain Lascoux},
title = {{Polynômes de Kazhdan-Lusztig pour les variétés de Schubert vexillaires}},
year = {1995},
journal = {C. R. Acad. Sci. Paris Sér. I Math.},
volume = {321},
number = {6},
pages = {667--670}
}
Polynômes de Schubert: Une approche historique (1995, 15 pp.)
Alain Lascoux
We tell the story of Schubert Polynomials.
@article{lascoux1995polynomesschubert,
author = {Alain Lascoux},
title = {{Polynômes de Schubert: Une approche historique}},
year = {1995},
journal = {Discrete Math.},
volume = {139},
number = {1-3},
pages = {303--317},
doi = {10.1016/0012-365X(95)93984-D}
}
The LLPT Notes (1995, 142 pp.)
Dan Laksov, Alain Lascoux, Piotr Pragacz, Anders Thorup
@unpublished{laksov1995llptnotes,
author = {Dan Laksov and Alain Lascoux and Piotr Pragacz and Anders Thorup},
title = {{The LLPT Notes}},
year = {1995},
note = {Lecture notes, edited by A. Thorup, 1995 (with corrections and additions by D. Grinberg, 2017/18)}
}
Hecke Algebras at Roots of Unity and Crystal Bases of Quantum Affine Algebras (1996, 46 pp.)
Alain Lascoux, Bernard Leclerc, Jean-Yves Thibon
We present a fast algorithm for computing the global crystal basis of the basic 6^(sln)-module. This algorithm is based on combinatorial techniques which have been developed for dealing with modular representations of symmetric groups, and more generally with representations of Hecke algebras of type A at roots of unity. We conjecture that, upon specialization q —> 1, our algorithm computes the decomposition matrices of all Hecke algebras at a nth root of 1.
@article{lascoux1996heckealgebras,
author = {Alain Lascoux and Bernard Leclerc and Jean-Yves Thibon},
title = {{Hecke Algebras at Roots of Unity and Crystal Bases of Quantum Affine Algebras}},
year = {1996},
journal = {Comm. Math. Phys.},
volume = {181},
pages = {205--263},
doi = {10.1007/BF02101678}
}
Interpolation (1996, 40 pp.)
Alain Lascoux
@unpublished{lascoux1996interpolation,
author = {Alain Lascoux},
title = {{Interpolation}},
year = {1996},
note = {Unpublished}
}
Notes on Interpolation (1996, 51 pp.)
Alain Lascoux
A version of Interpolation for a summer school in Tianjin, China in June 1996. Includes exercises.
@unpublished{lascoux1996notesinterpolation,
author = {Alain Lascoux},
title = {{Notes on Interpolation}},
year = {1996},
note = {Unpublished}
}
Potentiel Yin sur le groupe symétrique (1996, 12 pp.)
Alain Lascoux
J'aurais tant aimé pouvoir cosigner cet article avec M.P. Schützenberger. Il n'est plus parmi nous, mais a pu voir toutefois la première version de ce texte qui rapporte le travail accompli dans sa dernière semaine.
@unpublished{lascoux1996potentielyin,
author = {Alain Lascoux},
title = {{Potentiel Yin sur le groupe symétrique}},
year = {1996},
note = {Unpublished}
}
Treillis et bases des groupes de Coxeter (1996, 38 pp.)
Alain Lascoux, Marcel-Paul Schützenberger
@article{lascoux1996treillisbases,
author = {Alain Lascoux and Marcel-Paul Schützenberger},
title = {{Treillis et bases des groupes de Coxeter}},
year = {1996},
journal = {Electron. J. Combin.},
volume = {3},
number = {2},
note = {Research Paper 27, the Foata Festschrift},
doi = {10.37236/1285}
}
Twisted Action of the Symmetric Group on the Cohomology of a Flag Manifold (1996, 15 pp.)
Alain Lascoux, Bernard Leclerc, Jean-Yves Thibon
Classes dual to Schubert cycles constitute a basis on the cohomology ring of the ag manifold , self-adjoint up to indexation with respect to the intersection form. F
@article{lascoux1996twistedaction,
author = {Alain Lascoux and Bernard Leclerc and Jean-Yves Thibon},
title = {{Twisted Action of the Symmetric Group on the Cohomology of a Flag Manifold}},
year = {1996},
journal = {Banach Center Publ.},
volume = {36},
number = {1},
pages = {111--124},
doi = {10.4064/-36-1-111-124}
}
Algebraic Combinatorics with MAPLE and ACE (c. 1997, 7 pp.)
Daniel Krob, Alain Lascoux, Bernard Leclerc, Jean-Yves Thibon, B.C.V. Ung, Sébastien Veigneau
@article{krob1997algebraiccombinatorics,
author = {Daniel Krob and Alain Lascoux and Bernard Leclerc and Jean-Yves Thibon and B.C.V. Ung and Sébastien Veigneau},
title = {{Algebraic Combinatorics with MAPLE and ACE}},
year = {1997},
journal = {Maple Tech. Newsl.},
volume = {4},
number = {1},
pages = {43--50}
}
Flag Varieties and the Yang-Baxter Equation (1997, 14 pp.)
Alain Lascoux, Bernard Leclerc, Jean-Yves Thibon
We investigate certain bases of Hecke algebras defined by means of the Yang- Baxter equation, which we call Yang-Baxter bases. These bases are essentially self- adjoint with respect to a canonical bilinear form. In the case of the degenerate Hecke algebra, we identify the coefficients in the expansion of the Yang-Baxter basis on the usual basis of the algebra with specializations of double Schubert polynomials. We also describe the expansions associated to other specializations of the generic Hecke algebra.
@article{lascoux1997flagvarieties,
author = {Alain Lascoux and Bernard Leclerc and Jean-Yves Thibon},
title = {{Flag Varieties and the Yang-Baxter Equation}},
year = {1997},
journal = {Lett. Math. Phys.},
volume = {40},
number = {1},
pages = {75--90},
doi = {10.1023/A:1007307826670}
}
Ribbon Tableaux, Hall-Littlewood Functions, Quantum Affine Algebras and Unipotent Varieties (1997, 35 pp.)
Alain Lascoux, Bernard Leclerc, Jean-Yves Thibon
We introduce a new family of symmetric functions, which are q-analogues of products of Schur functions, defined in terms of ribbon tableaux. These functions can be interpreted in terms of the Fock space representa- b ), and are related to Hall-Littlewood functions via the geom- tion F of Uq (sl n etry of flag varieties. We present a series of conjectures, and prove them in special cases. The essential step in proving that these functions are actually symmetric consists in the calculation of a basis of highest weight vectors of F using ribbon tableaux.
@article{lascoux1997ribbontableaux,
author = {Alain Lascoux and Bernard Leclerc and Jean-Yves Thibon},
title = {{Ribbon Tableaux, Hall-Littlewood Functions, Quantum Affine Algebras and Unipotent Varieties}},
year = {1997},
journal = {J. Math. Phys.},
volume = {38},
number = {2},
pages = {1041--1068},
doi = {10.1063/1.531807}
}
Young's Natural Idempotents as Polynomials (1997, 9 pp.)
Alain Lascoux
Coding permutations as monomials, one obtains a compact expression of representa- tives of Young's natural idempotents for the symmetric group, or of q-idempotents in the Hecke algebra.
@article{lascoux1997youngs,
author = {Alain Lascoux},
title = {{Young's Natural Idempotents as Polynomials}},
year = {1997},
journal = {Ann. Comb.},
volume = {1},
pages = {91--98},
doi = {10.1007/BF02558465}
}
ACE: User's Reference Manual, Version 3.0 (book) (1998, 448 pp.)
Sébastien Veigneau, Alain Lascoux
@unpublished{veigneau1998aceuser,
author = {Sébastien Veigneau and Alain Lascoux},
title = {{ACE: User's Reference Manual, Version 3.0 (book)}},
year = {1998},
note = {Unpublished}
}
Determinantal expressions for Macdonald polynomials (1998, 21 pp.)
Luc Lapointe, Alain Lascoux, Jennifer Morse
We show that the action of classical operators associated to the Mac- donald polynomials on the basis of Schur functions, Sλ [X(t − 1)/(q − 1)], can be reduced to addition in λ−rings. This provides explicit formulas for the Macdonald polynomials expanded in this basis as well as in the ordinary Schur basis, Sλ [X], and the monomial basis, mλ [X].
@article{lapointe1998determinantalexpressions,
author = {Luc Lapointe and Alain Lascoux and Jennifer Morse},
title = {{Determinantal expressions for Macdonald polynomials}},
year = {1998},
journal = {Internat. Math. Res. Notices},
volume = {1998},
number = {18},
pages = {957--978},
doi = {10.1155/S1073792898000579}
}
Operator Calculus for Q-Polynomials and Schubert Polynomials (1998, 38 pp.)
Alain Lascoux, Piotr Pragacz
@article{lascoux1998operatorcalculus,
author = {Alain Lascoux and Piotr Pragacz},
title = {{Operator Calculus for Q-Polynomials and Schubert Polynomials}},
year = {1998},
journal = {Adv. Math.},
volume = {140},
pages = {143--185},
note = {Article No. AI981757},
doi = {10.1006/aima.1998.1757}
}
Operators on Polynomials (c. 1998, 85 pp.)
Alain Lascoux
@unpublished{lascoux1998operatorspolynomials,
author = {Alain Lascoux},
title = {{Operators on Polynomials}},
year = {1998},
note = {Unpublished}
}
Ordonner le Groupe Symétrique: Pourquoi Utiliser l'Algèbre de Iwahori-Hecke ? (1998, 10 pp.)
Alain Lascoux
The Bruhat order on the symmetric group is de ned by means of subwords of reduced decompositions of permutations as prod- ucts of simple transpositions. Ehresmann gave a di erent description by considering any permutation as a chain of sets and comparing component- wise the chains. A third method reduces the Bruhat order to the inclusion order on sets, by associating to any permutation a set of bigrassmannian permutations. This amounts to embed the symmetric group into a lat- tice which is distributive. The last manner to understand the Bruhat order is to use a distinguished linear basis of the Iwahori-Hecke algebra of the symmetric group, and this involves computing polynomials due to Kazhdan & Lusztig; we explicit these polynomials in the case of vexillary permutations. 1991 Mathematics Subject Classi cation: 05E10, 20C30
@article{lascoux1998ordonnergroupe,
author = {Alain Lascoux},
title = {{Ordonner le Groupe Symétrique: Pourquoi Utiliser l'Algèbre de Iwahori-Hecke ?}},
year = {1998},
journal = {Doc. Math.},
pages = {355--364},
note = {Extra Vol. ICM Berlin 1998, vol. III},
doi = {10.4171/dms/1-3/35}
}
Talk Slides Archive (1998–2008) (c. 1998)
Alain Lascoux
This archive contains the source and, where available, compiled files for miscellaneous talk slides by Alain Lascoux (circa 1998–2008), extracted from his personal working archive. Files included:
lascoux__talk_1998_divided_differences_and_macdonald_polynomials.tex
lascoux__talk_1999_gian_carlo_rota_and_addition_of_alphabets.tex
lascoux__talk_2000_calculs_multivaries__ExpIGM.pdf
lascoux__talk_2000_calculs_multivaries__ExpIGM.ps
lascoux__talk_2000_computer_algebra_demonstration_dec2000.tex
lascoux__talk_2002_interp_newton_poly_Schub_eqns_YB__SchubertAtLacanau.pdf
lascoux__talk_2002_interp_newton_poly_Schub_eqns_YB__SchubertAtLacanau.ps
lascoux__talk_2002_interp_newton_poly_Schub_eqns_YB__SchubertAtLacanau.tex
lascoux__talk_2003_20_ans_Schub_et_Groth__GrothLothSlides.pdf
lascoux__talk_2003_20_ans_Schub_et_Groth__GrothLothSlides.ps
lascoux__talk_2003_20_ans_Schub_et_Groth__GrothLothSlides.tex
lascoux__talk_2003_computer_algebra_demonstration_mar2003.tex
lascoux__talk_2003_importance_of_2_in_algebraic_combinatorics.tex
lascoux__talk_2003_square_ice_and_alternating_sign_matrices.tex
lascoux__talk_2004_addition_of_minusplus_1.tex
lascoux__talk_2006_computer_algebra_demonstration_jan2006.tex
lascoux__talk_2007_calculs_algebrique_en_plusieurs_variables.tex
lascoux__talk_2007_gaudin_functions_of_arbitrary_level__Quebec07.pdf
lascoux__talk_2007_subtraction_of_0_and_frobenius_formula.tex
lascoux__talk_2008_schubert_and_macdonald_functions_a_parallel__MsriExp.pdf
lascoux__talk_bonne_fete_claudio.tex
lascoux__talk_cores_and_kazhdan_lusztig_kerber_fest.tex
lascoux__talk_kazhdan_lusztig_to_demazure.tex
lascoux__talk_ordering_the_symmetric_group.tex
lascoux__talk_representations_of_the_symmetric_group.tex
lascoux__talk_square_ice_enumeration_basilicata.tex
About the "y" in the χy-characteristic of Hirzebruch (1999, 12 pp.)
Alain Lascoux
We indicate how Hecke algebras, Yang-Baxter equation, Hall- Littlewood polynomials, Macdonald polynomials, are related to the parameter y that Hirzebruch introduced in his study of Riemann-Roch theorem.
@incollection{lascoux1999abouty,
author = {Alain Lascoux},
title = {{About the "y" in the χy-characteristic of Hirzebruch}},
year = {1999},
booktitle = {Algebraic Geometry: Hirzebruch 70 (Warsaw, 1998)},
volume = {241},
pages = {285--296},
publisher = {Amer. Math. Soc., Providence, RI},
series = {Contemp. Math.},
doi = {10.1090/conm/241/03640}
}
Polynômes harmoniques (1999, 9 pp.)
Alain Lascoux
Nous décrivons l'espace des polynômes annulés par le Laplacien ½ Σ d²/dx² en tant que représentation du groupe symétrique.
@unpublished{lascoux1999polynomesharmoniques,
author = {Alain Lascoux},
title = {{Polynômes harmoniques}},
year = {1999},
note = {Unpublished}
}
Square-Ice Enumeration (1999, 15 pp.)
Alain Lascoux
Starting with plane partitions possessing certain type of symmetries, many combinatorial objects came to the fore, the enumeration of which was the subject of intensive studies during the last twenty years, with of course, seminal contributions of George Andrews. Thanks to a detour through two-dimensional ice models, algebraic computations cristallised to the description of a certain determinant of Cauchy type. Dividing this determinant by some straightforward factors, one is reduced to studying a symmetric polynomial in two sets of variables. We show how to separate the variables with the help of divided differences, and obtain the desired symmetric function as a product of two rectangular matrices, each of them involving only one set of variables. In the same run, we reduce the dimension by 1 and factorize the determinant associated to the Bethe model of a 1-dimensional gas of bosons.
@article{lascoux1999squareice,
author = {Alain Lascoux},
title = {{Square-Ice Enumeration}},
year = {1999},
journal = {Sém. Lothar. Combin.},
volume = {42},
note = {Article B42; dedicated to George Andrews on his sixtieth birthday}
}
Determinantal expression and recursion for Jack polynomials (2000, 7 pp.)
Luc Lapointe, Alain Lascoux, Jennifer Morse
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 −(rα + s), with r and s ∈ ℕ⁺, the entries above the sub-diagonal are nonnegative 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.
@article{lapointe2000determinantalexpression,
author = {Luc Lapointe and Alain Lascoux and Jennifer Morse},
title = {{Determinantal expression and recursion for Jack polynomials}},
year = {2000},
journal = {Electron. J. Combin.},
volume = {7},
note = {#N1},
doi = {10.37236/1539}
}
Factorization of Kazhdan–Lusztig elements for Grassmanians (2000, 9 pp.)
Alexander Kirillov, Jr., Alain Lascoux
We show that the Kazhdan-Lusztig basis elements Cw of the Hecke algebra of the symmetric group, when w ∈ Sn corresponds to a Schubert subvariety of a Grassmann variety, can be written as a product of factors of the form Ti + fj (v), where fj are rational functions.
@incollection{kirillov2000factorizationkazhdan,
author = {Alexander Kirillov, Jr. and Alain Lascoux},
title = {{Factorization of Kazhdan–Lusztig elements for Grassmanians}},
year = {2000},
booktitle = {Combinatorial Methods in Representation Theory},
volume = {28},
pages = {143--154},
publisher = {Kinokuniya, Tokyo},
series = {Adv. Stud. Pure Math.},
eprint = {math/9902072},
archivePrefix = {arXiv},
doi = {10.2969/aspm/02810143}
}
Factorizations in Schubert Cells (2000, 30 pp.)
Christian Kassel, Alain Lascoux, Christophe Reutenauer
For any reduced decomposition i = (i1 , i2 , . . . , iN ) of a permutation w and any ring R we construct a bijection
@article{kassel2000factorizationsschubert,
author = {Christian Kassel and Alain Lascoux and Christophe Reutenauer},
title = {{Factorizations in Schubert Cells}},
year = {2000},
journal = {Adv. Math.},
volume = {150},
pages = {135--169},
note = {doi:10.1006/aima.1999.1862},
doi = {10.1006/aima.1999.1862}
}
Motzkin paths and powers of continued fractions (2000, 4 pp.)
Alain Lascoux
We show that the combinatorial description of cumulants by Lehner, in terms of Motzkin paths, can be extended to the description of powers of continued fractions.
@article{lascoux2000motzkinpaths,
author = {Alain Lascoux},
title = {{Motzkin paths and powers of continued fractions}},
year = {2000},
journal = {Sém. Lothar. Combin.},
volume = {44},
note = {Article B44e}
}
Orthogonal Divided Differences and Schubert Polynomials, P̃-Functions, and Vertex Operators (2000, 26 pp.)
Alain Lascoux, Piotr Pragacz
Using divided di erences associated with the orthogonal groups, we investigate the structure of the polynomial rings over the rings of invariants of the corresponding Weyl groups. We study in more detail the action of orthogonal divided di erences on some distinguished symmetric polynomials (Pe - polynomials) and relate it to vertex operators. Relevant families of orthogonal Schubert polynomials, generalizing Pe-polynomials, and well-suited to intersection theory computations, are also studied.
@article{lascoux2000orthogonaldivided,
author = {Alain Lascoux and Piotr Pragacz},
title = {{Orthogonal Divided Differences and Schubert Polynomials, P̃-Functions, and Vertex Operators}},
year = {2000},
journal = {Michigan Math. J.},
volume = {48},
doi = {10.1307/mmj/1030132727}
}
Pour le monoïde plaxique (c. 2000, 7 pp.)
Marcel-Paul Schützenberger
@unpublished{schutzenberger2000pourmonoide,
author = {Marcel-Paul Schützenberger},
title = {{Pour le monoïde plaxique}},
year = {2000},
note = {Unpublished}
}
Schur Q-Functions and Degeneracy Locus Formulas for Morphisms with Symmetries (2000, 21 pp.)
Alain Lascoux, Piotr Pragacz
We give closed-form formulas for the fundamental classes of degeneracy loci as- sociated with vector bundle maps given locally by (not necessary square) matrices which are symmetric (resp. skew-symmetric) w.r.t. the main diagonal. Our description uses essentially Schur Q-polynomials of a bundle and is based on a push-forward formula for these polynomials in a Grassmann bundle, established in [P4].
@incollection{lascoux2000schurq,
author = {Alain Lascoux and Piotr Pragacz},
title = {{Schur Q-Functions and Degeneracy Locus Formulas for Morphisms with Symmetries}},
year = {2000},
booktitle = {Recent Progress in Intersection Theory},
pages = {239--263},
publisher = {Birkhäuser Boston},
series = {Trends Math.},
eprint = {math/9901126},
archivePrefix = {arXiv},
doi = {10.1007/978-1-4612-1316-1_8}
}
Une identité remarquable en théorie des partitions (2000, 16 pp.)
Alain Lascoux, Michel Lassalle
We prove an identity about partitions, previously conjectured in the study of shifted Jack polynomials. The proof given is using λ-ring techniques. It would be interesting to obtain a bijective proof.
@article{lascoux2000identiteremarquable,
author = {Alain Lascoux and Michel Lassalle},
title = {{Une identité remarquable en théorie des partitions}},
year = {2000},
journal = {Math. Ann.},
volume = {318},
pages = {299--313},
doi = {10.1007/s002080000121}
}
Youngs’s Representations of the Symmetric Group (2000, 11 pp.)
Alain Lascoux
We describe the different matrices, due to Young, representing the symmetric group, by reading the same graph with various labellings. Orthogonal idempotents are obtained in the like manner. The only mathematical tools needed for these constructions are comparison of integers and addition of vectors.
@unpublished{lascoux2000youngss,
author = {Alain Lascoux},
title = {{Youngs’s Representations of the Symmetric Group}},
year = {2000},
note = {Unpublished}
}
Youngs’s Representations of the Symmetric Group (2000, 11 pp.)
Alain Lascoux
We describe the different matrices, due to Young, representing the symmetric group, by reading the same graph with various labellings. Orthogonal idempotents are obtained in the like manner. The only mathematical tools needed for these constructions are comparison of integers and addition of vectors.
@unpublished{lascoux2000youngssb,
author = {Alain Lascoux},
title = {{Youngs’s Representations of the Symmetric Group}},
year = {2000},
note = {Unpublished}
}
About division by 1 (2001, 7 pp.)
Alain Lascoux
The Euclidean division of two formal series in one variable produces a sequence of series that we obtain explicitly, remarking that the case where one of the two initial series is 1 is sufficiently generic. As an application, we define a Wronskian of symmetric functions.
@article{lascoux2001aboutdivision,
author = {Alain Lascoux},
title = {{About division by 1}},
year = {2001},
journal = {Electron. J. Combin.},
volume = {8},
number = {1},
note = {#N8},
doi = {10.37236/1593}
}
Alphabet Splitting (c. 2001, 16 pp.)
Alain Lascoux
We stress the importance of addition (of alphabets) in the mathematical work of Gian-Carlo Rota, in particular as concerns his ”Finite Operator Calculus”.
@unpublished{lascoux2001alphabetsplitting,
author = {Alain Lascoux},
title = {{Alphabet Splitting}},
year = {2001},
note = {Unpublished}
}
Chern and Yang through Ice (c. 2001, 17 pp.)
Alain Lascoux
Characteristic classes for flags of vector bundles and Yang-Baxter coefficients are related to the flag variety for the linear group, and, ultimately, to the Ehresmann-Bruhat order on the symmetric group. This order can be interpreted in terms of an embedding of the symmetric group into the lattice of alternating-sign matrices (in bijection with square ice configurations). By decomposing the set of ice configurations into cells indexed by permutations, we are able to explicit characteristic classes, Grothendieck polynomials and Yang-Baxter coefficients from a simple weight on ice configurations.
@unpublished{lascoux2001chernyang,
author = {Alain Lascoux},
title = {{Chern and Yang through Ice}},
year = {2001},
note = {Unpublished}
}
Couper les alphabets en quatre (c. 2001, 16 pp.)
Alain Lascoux
We stress the importance of addition (of alphabets) in the mathematical work of Gian-Carlo Rota, in particular for what concerns his ”Finite Operator Calculus”.
@unpublished{lascoux2001couperalphabets,
author = {Alain Lascoux},
title = {{Couper les alphabets en quatre}},
year = {2001},
note = {Unpublished}
}
Fonctions symétriques (2001, 17 pp.)
Alain Lascoux
@article{lascoux2001fonctionssymetriques,
author = {Alain Lascoux},
title = {{Fonctions symétriques}},
year = {2001},
journal = {Publ. I.R.M.A. Strasbourg, Actes 8e Séminaire Lotharingien},
pages = {37--53}
}
Noncommutative symmetric functions and quasi-symmetric functions with two and more parameters (2001, 14 pp.)
Florent Hivert, Alain Lascoux, Jean-Yves Thibon
We define two-parameter families of noncommutative symmetric functions and quasi-symmetric functions, which appear to be the proper analogues of the Macdonald symmetric functions in these settings.
@article{hivert2001noncommutativesymmetric,
author = {Florent Hivert and Alain Lascoux and Jean-Yves Thibon},
title = {{Noncommutative symmetric functions and quasi-symmetric functions with two and more parameters}},
year = {2001},
eprint = {math/0106191},
archivePrefix = {arXiv}
}
Ordering the affine symmetric group (2001, 13 pp.)
Alain Lascoux
We review several descriptions of the affine symmetric group. We explicit the basis of its Bruhat order.
@incollection{lascoux2001orderingaffine,
author = {Alain Lascoux},
title = {{Ordering the affine symmetric group}},
year = {2001},
booktitle = {Algebraic Combinatorics and Applications (Gößweinstein, 1999)},
pages = {219--231},
publisher = {Springer, Berlin},
doi = {10.1007/978-3-642-59448-9_15}
}
Transition on Grothendieck Polynomials (2001, 16 pp.)
Alain Lascoux
Grothendieck polynomials represent Schubert varieties in the Grothendieck ring of the flag manifold (for Gl(n, ℂ)). We describe how general Grothendieck polynomials are related to those for Grassmann manifolds, which themselves are deformations of Schur functions.
@incollection{lascoux2001transitiongrothendieck,
author = {Alain Lascoux},
title = {{Transition on Grothendieck Polynomials}},
year = {2001},
booktitle = {Physics and Combinatorics 2000 (Nagoya, 2000)},
pages = {164--179},
publisher = {World Scientific, River Edge, NJ},
doi = {10.1142/9789812810007_0007}
}
Yang-Baxter Graphs, Jack and Macdonald Polynomials (2001, 33 pp.)
Alain Lascoux
We describe properties of the affine graph underlying the recursions between the dif- ferent varieties of nonsymmetric Macdonald and Jack polynomials. We use an arbitrary function of one variable in the definition of affine edges, and of Cherednik’s elements, to unify the dif- ferent theories. We describe the symmetrizing operators furnishing the symmetric polynomials from the nonsymmetric ones.
@article{lascoux2001yangbaxter,
author = {Alain Lascoux},
title = {{Yang-Baxter Graphs, Jack and Macdonald Polynomials}},
year = {2001},
journal = {Ann. Comb.},
volume = {5},
pages = {397--424},
doi = {10.1007/s00026-001-8019-3}
}
Algebraic Combinatorics on Words (book) (2002, 455 pp.)
M. Lothaire
@book{lothaire2002algebraic,
title={Algebraic combinatorics on words},
author={Lothaire, Monsieur},
volume={90},
year={2002},
publisher={Cambridge university press},
doi = {10.1017/cbo9781107326019}
}
Jacobians of Symmetric Polynomials (2002, 4 pp.)
Alain Lascoux, Piotr Pragacz
We give the Jacobian of any family of complete symmetric functions, or of power sums, in a finite number of variables.
@article{lascoux2002jacobianssymmetric,
author = {Alain Lascoux and Piotr Pragacz},
title = {{Jacobians of Symmetric Polynomials}},
year = {2002},
journal = {Ann. Comb.},
volume = {6},
pages = {169--172},
doi = {10.1007/PL00012583}
}
Symmetric Functions (2002, 131 pp.)
Alain Lascoux
@unpublished{lascoux2002symmetricfunctions,
author = {Alain Lascoux},
title = {{Symmetric Functions}},
year = {2002},
note = {Unpublished}
}
The Plactic Monoid (Chapter 6, preliminary version) (2002, 29 pp.)
Alain Lascoux, Bernard Leclerc, Jean-Yves Thibon
@incollection{lascoux2002placticmonoid,
author = {Alain Lascoux and Bernard Leclerc and Jean-Yves Thibon},
title = {{The Plactic Monoid (Chapter 6, preliminary version)}},
year = {2002},
booktitle = {Algebraic Combinatorics on Words},
pages = {164--196},
publisher = {Cambridge Univ. Press, Cambridge},
editor = {M. Lothaire},
note = {Chapter 5, "The Plactic Monoid"},
doi = {10.1017/CBO9781107326019.006}
}
The Symmetric Group (2002, 81 pp.)
Alain Lascoux
@unpublished{lascoux2002symmetricgroup,
author = {Alain Lascoux},
title = {{The Symmetric Group}},
year = {2002},
note = {Unpublished}
}
q-identities from Lagrange and Newton interpolation (2002, 6 pp.)
Amy M. Fu, Alain Lascoux
Combining Newton and Lagrange interpolation, we give q-identities which generalize results of Van Hamme, Uchimura, Dilcher, and Prodinger.
@article{fu2002qidentities,
author = {Amy M. Fu and Alain Lascoux},
title = {{q-identities from Lagrange and Newton interpolation}},
year = {2002},
journal = {Adv. in Appl. Math.},
doi = {10.1016/S0196-8858(03)00024-1}
}
An involution for the Gauss identity (2003, 14 pp.)
William Y.C. Chen, Qing-Hu Hou, Alain Lascoux
We present an involution for a classical identity on the alternate sum of the Gauss coefficients in terms of the traditional Ferrers diagram. It turns out that the refinement of our involution with restrictions on the height of Ferrers diagram implies a generalization of the Gauss identity, which is a terminating form of the q-Kummer identity. Furthermore, we extend the Gauss identity to the pth root of unity.
@article{chen2003involutiongauss,
author = {William Y.C. Chen and Qing-Hu Hou and Alain Lascoux},
title = {{An involution for the Gauss identity}},
year = {2003},
journal = {J. Combin. Theory Ser. A},
doi = {10.1016/S0097-3165(03)00043-8}
}
Double Crystal Graphs (2003, 22 pp.)
Alain Lascoux
We show how to expand a non-symmetric Cauchy kernel ∏(1 − x_i y_j)^{-1} in the basis of Demazure characters for type A_n. The construction involves using the left and right structure of crystal graphs on words, and mostly reduces to properties of the jeu de taquin.
@incollection{lascoux2003doublecrystal,
author = {Alain Lascoux},
title = {{Double Crystal Graphs}},
year = {2003},
booktitle = {Studies in Memory of Issai Schur},
volume = {210},
pages = {95--114},
publisher = {Birkhäuser Boston},
series = {Progr. Math.},
doi = {10.1007/978-1-4612-0045-1_5}
}
Double Sylvester sums for subresultants and multi-Schur functions (2003, 25 pp.)
Alain Lascoux, Piotr Pragacz
Sylvester has announced formulas expressing the subresultants (or the successive polynomial remainders for the Euclidean division) of two polynomials, in terms of some double sums over the roots of the two polynomials. We prove Sylvester formulas using the techniques of multivariate polynomials involving multi-Schur functions and divided differences.
@article{lascoux2003doublesylvester,
author = {Alain Lascoux and Piotr Pragacz},
title = {{Double Sylvester sums for subresultants and multi-Schur functions}},
year = {2003},
journal = {J. Symbolic Comput.},
volume = {35},
pages = {689--710},
doi = {10.1016/S0747-7171(03)00038-5}
}
Euclid and Orthogonal Polynomials (2003, 6 pp.)
Alain Lascoux
In the classical theory of elimination, or of Sturm sequences, one encounters relations which are better understood when translated in terms of orthogonal polynomials. Conversely, one may safely associate to the Tchebychef-Christoffel-Darboux kernel the names of Bézout, Sylvester, Hermite, Brioschi, and recognize that these mathematicians treated the case of a finite discrete measure, with the help of the Euclidean algorithm and of the Lagrange interpolation.
@unpublished{lascoux2003euclidorthogonal,
author = {Alain Lascoux},
title = {{Euclid and Orthogonal Polynomials}},
year = {2003},
note = {Unpublished}
}
Littlewood's formulas for characters of orthogonal and symplectic groups (2003, 10 pp.)
Alain Lascoux
Littlewood gave expansions of products of the type ∏ 1/(1 − a_i a_j). Several authors published generalizations involving a small number of extra factors ∏ 1/(1 − x a_i). We show that the method of Littlewood (that we express in terms of λ-rings) covers these extensions. We also express Littlewood's coefficients in terms of orthogonal or symplectic Schur functions.
@incollection{lascoux2003littlewoods,
author = {Alain Lascoux},
title = {{Littlewood's formulas for characters of orthogonal and symplectic groups}},
year = {2003},
booktitle = {Algebraic Combinatorics and Quantum Groups},
pages = {125--133},
publisher = {World Scientific, River Edge, NJ},
editor = {N. Jing},
doi = {10.1142/9789812775405_0006}
}
Schubert & Grothendieck : un Bilan Bidécennal (2003, 30 pp.)
Alain Lascoux
We give a dozen formulas concerning Schubert and Grothen- dieck polynomials, and their interrelations, half of them being new, and most of them interesting. In particular, we describe explicitly the de- composition of Schubert polynomials as positive sums of Grothendieck polynomials, and we show that non-commutative Schubert polynomials are obtained by reading the columns of a two-dimensional Cauchy kernel. A six pages summary in English has been added.
@article{lascoux2003schubertgrothendieck,
author = {Alain Lascoux},
title = {{Schubert \& Grothendieck : un Bilan Bidécennal}},
year = {2003},
journal = {Sém. Lothar. Combin.},
volume = {50},
note = {Article B50i, 32 pp.}
}
Symmetric Functions and Combinatorial Operators on Polynomials (2003, 96 pp.)
Alain Lascoux
We present notes about symmetric functions and their generalizations.
@book{lascoux2003symmetricfunctions,
author = {Alain Lascoux},
title = {{Symmetric Functions and Combinatorial Operators on Polynomials}},
year = {2003},
number = {99},
publisher = {American Mathematical Society, Providence, RI},
series = {CBMS Regional Conference Series in Mathematics},
doi = {10.1090/cbms/099}
}
Tableau Atoms and a New Macdonald Positivity Conjecture (2003, 40 pp.)
Luc Lapointe, Alain Lascoux, Jennifer Morse
Let 3 be the space of symmetric functions, and let Vk be the subspace spanned by the modified Schur functions {Sλ [X/(1−t)]}λ1 ≤k . We introduce a new family of symmetric (k) polynomials, {Aλ [X ; t]}λ1 ≤k , constructed from sums of tableaux using the charge (k) statistic. We conjecture that the polynomials Aλ [X ; t] form a basis for Vk 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 also substantially refine it. (k) Our construction of the Aλ [X ; t] relies on the use of tableau combinatorics and yields various properties and conjectures on the nature of these polynomials. Another (k) important development following from our investigation is that the Aλ [X ; t] seem to play the same role for Vk as the Schur functions do for 3. In particular, this has led us to the discovery of many generalizations of properties held by the Schur functions, such as Pieri-type and Littlewood-Richardson-type coefficients.
@article{lapointe2003tableauatoms,
author = {Luc Lapointe and Alain Lascoux and Jennifer Morse},
title = {{Tableau Atoms and a New Macdonald Positivity Conjecture}},
year = {2003},
journal = {Duke Math. J.},
volume = {116},
number = {1},
pages = {103--146},
doi = {10.1215/s0012-7094-03-11614-2}
}
The Newton interpolation formula, with more variables (c. 2003, 6 pp.)
Alain Lascoux
@unpublished{lascoux2003newtoninterpolation,
author = {Alain Lascoux},
title = {{The Newton interpolation formula, with more variables}},
year = {2003},
note = {Unpublished}
}
The singular locus of a Schubert variety (2003, 37 pp.)
Christian Kassel, Alain Lascoux, Christophe Reutenauer
The singular locus of a Schubert variety Xµ in the flag variety for GLn (C) is the union of Schubert varieties Xν , where ν runs over a set Sg(µ) of permutations in Sn . We describe completely the maximal elements of Sg(µ) under the Bruhat order, thus determining the irreducible components of the singular locus of Xµ .
@article{kassel2003singularlocus,
author = {Christian Kassel and Alain Lascoux and Christophe Reutenauer},
title = {{The singular locus of a Schubert variety}},
year = {2003},
journal = {J. Algebra},
volume = {269},
pages = {74--108},
doi = {10.1016/S0021-8693(03)00014-0}
}
Addition of ±1 : Application to Arithmetic (2004, 9 pp.)
Alain Lascoux
We show that some classical identities about multiplicative func- tions, and the Riemann zeta-function, may conveniently be interpreted in terms of the addition of ±1 to some alphabets. Résumé Nous montrons que des identités classiques concernant des fonc- tions multiplicatives de la théorie des nombres, et la fonction zeta de Riemann, s’interprètent commodément en terme de l’opération : “Ajouter ±1 à un alphabet”.
@article{lascoux2004addition1,
author = {Alain Lascoux},
title = {{Addition of ±1 : Application to Arithmetic}},
year = {2004},
journal = {Sém. Lothar. Combin.},
volume = {52},
note = {Article B52a}
}
Communication Sylvester’s bijection between strict and odd partitions (2004, 4 pp.)
Alain Lascoux
We show that Sylvester’s bijection between strict partitions and odd ones can be obtained by an appropriate coding of partitions.
@article{lascoux2004communicationsylvester,
author = {Alain Lascoux},
title = {{Communication Sylvester’s bijection between strict and odd partitions}},
year = {2004},
journal = {Discrete Math.},
volume = {277},
pages = {275--278},
doi = {10.1016/j.disc.2002.02.001}
}
Continued fractions for Rogers–Szegö polynomials (2004, 11 pp.)
Qing-Hu Hou, Alain Lascoux, Yan-Ping Mu
Lattice Graphs and Schubert Polynomials (2004, 15 pp.)
Alain Lascoux, Hui Lin, Arthur L. B. Yang
We define new planar combinatorial objects that we call lattice graphs. We generalize divided differences into operators, still satisfying braid relations, and operating on the space of lattice graphs. This allows us to interpret Schubert polynomials as sums of lattice graphs. By putting appropriate weights on lattice graphs, we recover the usual Schubert polynomials in two sets of variables.
@unpublished{lascoux2004latticegraphs,
author = {Alain Lascoux and Hui Lin and Arthur L. B. Yang},
title = {{Lattice Graphs and Schubert Polynomials}},
year = {2004},
note = {Unpublished}
}
Muir and the Theory of Recurrents (c. 2004, 6 pp.)
Alain Lascoux
@unpublished{lascoux2004muirtheory,
author = {Alain Lascoux},
title = {{Muir and the Theory of Recurrents}},
year = {2004},
note = {Unpublished}
}
Orthogonality Properties of Demazure Characters (2004, 16 pp.)
Alain Lascoux
@unpublished{lascoux2004orthogonalityproperties,
author = {Alain Lascoux},
title = {{Orthogonality Properties of Demazure Characters}},
year = {2004},
note = {Unpublished}
}
Partition analysis and symmetrizing operators (2004, 5 pp.)
Amy M. Fu, Alain Lascoux
Using a symmetrizing operator, we give a new expression for the Omega operator used by MacMa- hon in Partition Analysis, and given a new life by Andrews, Paule and Riese. Our result is stated in terms of Schur functions.
@article{fu2004partitionanalysis,
author = {Amy M. Fu and Alain Lascoux},
title = {{Partition analysis and symmetrizing operators}},
year = {2004},
journal = {J. Combin. Theory Ser. A},
doi = {10.1016/j.jcta.2004.08.005}
}
Pieri Formulas for Demazure Characters (2004, 11 pp.)
Alain Lascoux, Hui Lin, Arthur L. B. Yang
We give a Pieri formula for key polynomials (also called Demazure characters). It implies an explicit decomposition of Schubert polynomials in terms of key polynomials. The constructions extend to the non-commutative cases.
Schur Functions Series (2004, 18 pp.)
Alain Lascoux
@unpublished{lascoux2004schurfunctions,
author = {Alain Lascoux},
title = {{Schur Functions Series}},
year = {2004},
note = {Unpublished}
}
Vertex Operators and the Class Algebras of Symmetric Groups (2004, 23 pp.)
Alain Lascoux, Jean-Yves Thibon
We exhibit a vertex operator which implements multiplication by power-sums of Jucys-Murphy elements in the centers of the group al- gebras of all symmetric groups simultaneously. The coefficients of this operator generate a representation of W1+∞ , to which operators mul- tiplying by normalized conjugacy classes are also shown to belong. A new derivation of such operators based on matrix integrals is pro- posed, and our vertex operator is used to give an alternative approach to the polynomial functions on Young diagrams introduced by Kerov and Olshanski.
@article{lascoux2004vertexoperators,
author = {Alain Lascoux and Jean-Yves Thibon},
title = {{Vertex Operators and the Class Algebras of Symmetric Groups}},
year = {2004},
journal = {J. Math. Sci.},
volume = {121},
number = {3},
doi = {10.1023/b:joth.0000024619.77778.3d}
}
q-Identities related to overpartitions and divisor functions (2004, 6 pp.)
Amy M. Fu, Alain Lascoux
We generalize and prove two conjectures of Corteel and Lovejoy related to overpartitions and divisor functions.
Bezoutians, Euclidean Algorithm, and Orthogonal Polynomials (2005, 22 pp.)
Alain Lascoux, Piotr Pragacz
We prove a quadratic expression for the Bezoutian of two univariate polynomials in terms of the remainders for the Euclidean algorithm. In case of two polynomials of the same degree, or of consecutive degrees, this allows us to interpret their Bezoutian as the Christoffel- Darboux kernel for a finite family of orthogonal polynomials, arising from the Euclidean al- gorithm. We give orthogonality properties of remainders, and reproducing properties of Be- zoutians.
@article{lascoux2005bezoutianseuclidean,
author = {Alain Lascoux and Piotr Pragacz},
title = {{Bezoutians, Euclidean Algorithm, and Orthogonal Polynomials}},
year = {2005},
journal = {Ann. Comb.},
volume = {9},
pages = {301--319},
doi = {10.1007/s00026-005-0259-1}
}
Evaluation of some Hankel determinants (2005, 10 pp.)
Qing-Hu Hou, Alain Lascoux, Yan-Ping Mu
We evaluate some Hankel determinants of Meixner polynomials, associated to the series exp( α[i]zi / i), where [1], [2], . . . are the q-integers.
@article{hou2005evaluationhankel,
author = {Qing-Hu Hou and Alain Lascoux and Yan-Ping Mu},
title = {{Evaluation of some Hankel determinants}},
year = {2005},
journal = {Adv. in Appl. Math.},
volume = {34},
pages = {845--852},
doi = {10.1016/j.aam.2004.09.005}
}
Some composition determinants (2005, 9 pp.)
Josep Maria Brunat, Christian Krattenthaler, Alain Lascoux, Antonio Montes
We compute several parametric determinants in which rows and columns are indexed by compositions, where the entries are either products of binomial coefficients or products of powers. These results generalize previous determinant evaluations due to the first and fourth author [SIAM J. Matrix Anal. Appl. 23 (2001) 459–471] and [A polynomial generalization of the power-compositions determinant, Linear Multilinear Algebra, in press], and they prove two conjectures of the second author [Advanced determinant calculus: a complement, preliminary version].
@article{brunat2005compositiondeterminants,
author = {Josep Maria Brunat and Christian Krattenthaler and Alain Lascoux and Antonio Montes},
title = {{Some composition determinants}},
year = {2005},
journal = {Linear Algebra Appl.},
doi = {10.1016/j.laa.2005.11.013}
}
A question of Muir and Littlewood’s products (2006, 9 pp.)
Amy M. Fu, Alain Lascoux
a question of Muir, relating it to different determinantal expressions for the products i<j (y − We answer xi xj ) and i ⩽j (y − xi xj ), and for the products of these functions by an arbitrary Schur function.
@article{fu2006questionmuir,
author = {Amy M. Fu and Alain Lascoux},
title = {{A question of Muir and Littlewood’s products}},
year = {2006},
journal = {Linear Algebra Appl.},
doi = {10.1016/j.laa.2006.08.018}
}
Adding ±1 to the Argument of a Hall–littlewood Polynomial (2006, 17 pp.)
Alain Lascoux
@article{lascoux2006adding1,
author = {Alain Lascoux},
title = {{Adding ±1 to the Argument of a Hall–littlewood Polynomial}},
year = {2006},
journal = {Sém. Lothar. Combin.},
volume = {54},
note = {Article B54n, 17 pp.}
}
Non-Symmetric Hall-Littlewood Polynomials (2006, 15 pp.)
François Descouens, Alain Lascoux
Using the action of the Yang-Baxter elements of the Hecke algebra on polynomials, we define two bases of polynomials in n variables. The Hall-Littlewood polynomials are a subfamily of one of them. For q = 0, these bases specialize into the two families of classical Key polynomials (i.e. Demazure characters for type A). We give a scalar product for which the two bases are adjoint of each other.
@article{descouens2006nonsymmetric,
author = {François Descouens and Alain Lascoux},
title = {{Non-Symmetric Hall-Littlewood Polynomials}},
year = {2006},
journal = {Sém. Lothar. Combin.},
volume = {54},
note = {Article B54Ar}
}
Schubert and Macdonald Polynomials, a parallel (c. 2006, 25 pp.)
Alain Lascoux
@unpublished{lascoux2006schubertmacdonald,
author = {Alain Lascoux},
title = {{Schubert and Macdonald Polynomials, a parallel}},
year = {2006},
note = {Unpublished}
}
The Hecke algebra and structure constants of the ring of symmetric polynomials (2006, 25 pp.)
Alain Lascoux
We give half a dozen bases of the Hecke algebra of the symmetric group, and relate them to the basis of Geck-Rouquier, and to the basis of Jones, using matrices of change of bases of the ring of symmetric polynomials.
@article{lascoux2006heckealgebra,
author = {Alain Lascoux},
title = {{The Hecke algebra and structure constants of the ring of symmetric polynomials}},
year = {2006},
eprint = {math/0602379},
archivePrefix = {arXiv}
}
The differential equation satisfied by a plane curve of degree n (2006, 8 pp.)
Alain Lascoux
Eliminating the arbitrary coefficients in the equation of a generic plane curve of order n by computing sufficiently many derivatives, one obtains a differential equation. This is a projective invariant. The first one, corresponding to conics, has been obtained by Monge. Sylvester, Halphen, Cartan used invariants of higher order. The expression of these invariants is rather complicated, but becomes much simpler when interpreted in terms of symmetric functions.
@article{lascoux2006differentialequation,
author = {Alain Lascoux},
title = {{The differential equation satisfied by a plane curve of degree n}},
year = {2006},
journal = {Bull. Sci. Math.},
volume = {130},
pages = {354--359},
doi = {10.1016/j.bulsci.2006.02.002}
}
Gaudin functions, and Euler-Poincaré characteristics (2007, 14 pp.)
Alain Lascoux
Given two positive integers n, r, we define the Gaudin function of level r to be quotient of the numerator of det ((xi − yj )(xi − tyj ) · · · (xi − tr yj ))−1 i,j=1...n
@article{lascoux2007gaudinfunctions,
author = {Alain Lascoux},
title = {{Gaudin functions, and Euler-Poincaré characteristics}},
year = {2007},
eprint = {0709.1635},
archivePrefix = {arXiv}
}
Symmetric Functions & Combinatorial Operators on Polynomials (book) (c. 2007, 278 pp.)
Alain Lascoux
@unpublished{lascoux2007symmetricfunctions,
author = {Alain Lascoux},
title = {{Symmetric Functions \& Combinatorial Operators on Polynomials (book)}},
year = {2007},
note = {Unpublished}
}
The 6 Vertex Model and Schubert Polynomials (2007, 12 pp.)
Alain Lascoux
We enumerate staircases with fixed left and right columns. These objects cor- respond to ice-configurations, or alternating sign matrices, with fixed top and bottom parts. The resulting partition functions are equal, up to a normalization factor, to some Schubert polynomials.
@article{lascoux20076vertex,
author = {Alain Lascoux},
title = {{The 6 Vertex Model and Schubert Polynomials}},
year = {2007},
journal = {SIGMA},
volume = {3},
note = {Paper 029, 12 pp.},
doi = {10.3842/sigma.2007.029}
}
A Newton type rational interpolation formula (2008, 8 pp.)
Amy M. Fu, Alain Lascoux
We give a Newton type rational interpolation formula (Theorem 2.2). It contains as a special case the original Newton interpolation, as well as the interpolation formula of Liu, which allows to recover many important classical q-series identities. We show in particular that some bibasic identities are a consequence of our formula.
Rational Interpolation and Basic Hypergeometric Series (2008, 11 pp.)
Amy M. Fu, Alain Lascoux
We give a Newton type rational interpolation formula (Theorem 2.2). It contains as a special case the original Newton interpolation, as well as the recent interpolation formula of Zhi-Guo Liu, which allows to recover many important classical q-series identities. We show in particular that some bibasic identities are a consequence of our formula.
@article{fu2008rationalinterpolation,
author = {Amy M. Fu and Alain Lascoux},
title = {{Rational Interpolation and Basic Hypergeometric Series}},
year = {2008},
journal = {Adv. in Appl. Math.},
volume = {41},
number = {3},
pages = {452--458},
note = {Published under the title "A Newton type rational interpolation formula"},
doi = {10.1016/j.aam.2008.01.003}
}
Two Polynomial Identities (2008, 8 pp.)
Christian Krattenthaler, Alain Lascoux
We establish two polynomial identities. One of which generalizes a key identity in “A factorization theorem for classical group characters, with applications to plane partitions and rhombus tilings” by Mihai Ciucu and the first author [preprint 2008]. The other shows that the two sides in the identity can be expressed alternatively in form of a Pfaffian.
@unpublished{krattenthaler2008twopolynomial,
author = {Christian Krattenthaler and Alain Lascoux},
title = {{Two Polynomial Identities}},
year = {2008},
note = {Unpublished}
}
Non-symmetric Cauchy kernels for the classical groups (2009, 20 pp.)
Amy M. Fu, Alain Lascoux
We give non-symmetric versions of the Cauchy kernel and Littlewood’s kernels, corresponding to the types Aₙ, Bₙ, Cₙ and Dₙ, of the classical groups. We show that these new kernels are diagonal in the basis of two families of key polynomials (one of them being Demazure characters) obtained as images of dominant monomials under isobaric divided differences. We define scalar products such that the two families of key polynomials are adjoint to each other.
@article{fu2009nonsymmetric,
author = {Amy M. Fu and Alain Lascoux},
title = {{Non-symmetric Cauchy kernels for the classical groups}},
year = {2009},
journal = {J. Combin. Theory Ser. A},
volume = {116},
pages = {903--917},
doi = {10.1016/j.jcta.2008.10.007}
}
Nonsymmetric Interpolation Macdonald Polynomials and gln Basic Hypergeometric Series (2009, 30 pp.)
Alain Lascoux, Eric M. Rains, S. Ole Warnaar
The Knop–Sahi interpolation Macdonald polynomials are inhomogeneous and nonsymmetric generalisations of the well-known Macdonald polynomials. In this paper we apply the interpolation Macdonald polynomials to study a new type of basic hypergeometric series of type gln . Our main results include a new q-binomial theorem, a new q-Gauss sum, and several transformation formulae for gln series.
@article{lascoux2009nonsymmetricinterpolation,
author = {Alain Lascoux and Eric M. Rains and S. Ole Warnaar},
title = {{Nonsymmetric Interpolation Macdonald Polynomials and gln Basic Hypergeometric Series}},
year = {2009},
journal = {Transform. Groups},
volume = {14},
number = {3},
pages = {613--647},
doi = {10.1007/s00031-009-9061-1}
}
Pfaffians and Representations of the Symmetric Group (2009, 28 pp.)
Alain Lascoux
Pfaffians of matrices with entries z[i, j]/(xi + xj ), or determinants of matrices with entries z[i, j]/(xi − xj ), where the antisymmetrical indeterminates z[i, j] satisfy the Plücker relations, can be identified with a trace in an irreducible representation of a product of two symmetric groups. Using Young’s orthogonal bases, one can write explicit expressions of such Pfaffians and determinants, and recover in particular the evaluation of Pfaffians which appeared in the recent literature.
@article{lascoux2009pfaffiansrepresentations,
author = {Alain Lascoux},
title = {{Pfaffians and Representations of the Symmetric Group}},
year = {2009},
journal = {Acta Math. Sin. (Engl. Ser.)},
volume = {25},
number = {12},
pages = {1929--1950},
doi = {10.1007/s10114-009-6535-y}
}
Generalization of Scott’s Permanent Identity (2010, 5 pp.)
Alain Lascoux
Let x = {x1 , . . . , xr }, y = {y1 , . . . , yn }, z = {z1 , . . . , zn } be three sets of indeterminates. We give the value of the determinant Y (xy − z)−1 x∈x y∈y,z∈z when specializing y and z to the set of roots of y n − 1 and z n − ξ n , respectively.
@article{lascoux2010generalizationscott,
author = {Alain Lascoux},
title = {{Generalization of Scott’s Permanent Identity}},
year = {2010},
journal = {Sém. Lothar. Combin.},
volume = {61A},
note = {Article B61Ai}
}
Idempotents with polynomial coefficients (2010, 8 pp.)
Alain Lascoux
We combine Young idempotents in the group algebra of the sym- metric group with the action of the symmetric group on products of Vandermonde determinants to obtain idempotents with polynomial coefficients.
@article{lascoux2010idempotentspolynomial,
author = {Alain Lascoux},
title = {{Idempotents with polynomial coefficients}},
year = {2010},
eprint = {1011.5815},
archivePrefix = {arXiv}
}
Thom polynomials and Schur functions: the singularities A3(−) (2010, 18 pp.)
Alain Lascoux, Piotr Pragacz
Combining the “method of restriction equations” of Rimányi et al. with the techniques of symmetric functions, we establish the Schur function expansions of the Thom polynomials for the Morin singulari- ties A3 : (C• , 0) → (C•+k , 0) for any nonnegative integer k.
@article{lascoux2010thompolynomials,
author = {Alain Lascoux and Piotr Pragacz},
title = {{Thom polynomials and Schur functions: the singularities A3(−)}},
year = {2010},
journal = {Publ. Res. Inst. Math. Sci.},
volume = {46},
number = {1},
pages = {183--200},
doi = {10.2977/prims/5}
}
Branching Rules for Symmetric Macdonald Polynomials and sln Basic Hypergeometric Series (2011, 28 pp.)
Alain Lascoux, S. Ole Warnaar
A one-parameter generalisation Rλ (X; b) of the symmetric Mac- donald polynomials and interpolations Macdonald polynomials is studied from the point of view of branching rules. We establish a Pieri formula, evalua- tion symmetry, principal specialisation formula and q-difference equation for Rλ (X; b). We also prove a new multiple q-Gauss summation formula and several further results for sln basic hypergeometric series based on Rλ (X; b).
@article{lascoux2011branchingrules,
author = {Alain Lascoux and S. Ole Warnaar},
title = {{Branching Rules for Symmetric Macdonald Polynomials and sln Basic Hypergeometric Series}},
year = {2011},
journal = {Adv. in Appl. Math.},
volume = {46},
pages = {424--456},
doi = {10.1016/j.aam.2010.01.012}
}
Deformed Kazhdan–Lusztig elements and Macdonald polynomials (2012, 29 pp.)
Jan de Gier, Alain Lascoux, Mark Sorrell
We introduce deformations of Kazhdan-Lusztig elements and specialised nonsymmetric Macdon- ald polynomials, both of which form a distinguished basis of the polynomial representation of a maximal parabolic subalgebra of the Hecke algebra. We give explicit integral formula for these polynomials, and explicitly describe the transition matrices between classes of polynomials. We further develop a combinatorial interpretation of homogeneous evaluations using an expansion in terms of Schubert polynomials in the deformation parameters.
Hankel Pfaffians, Discriminants and Kazhdan-Lusztig bases (2012, 21 pp.)
Alain Lascoux
We use Kazhdan-Lusztig bases of representations of the symmetric group to express Pfaffians with entries (ai −aj )hi+j . In the case where the parameters ai are specialized to successive powers of q, and the hi are complete functions, we obtain the q-discriminant.
@article{lascoux2012hankelpfaffians,
author = {Alain Lascoux},
title = {{Hankel Pfaffians, Discriminants and Kazhdan-Lusztig bases}},
year = {2012},
journal = {J. Korean Math. Soc.},
volume = {49},
number = {4},
pages = {745--763},
doi = {10.4134/jkms.2012.49.4.745}
}
How many alphabets can a Schur function accommodate? (2012, 17 pp.)
Alain Lascoux
@article{lascoux2012howmany,
author = {Alain Lascoux},
title = {{How many alphabets can a Schur function accommodate?}},
year = {2012},
journal = {J. Combin.},
volume = {3},
number = {3},
pages = {591--607},
doi = {10.4310/joc.2012.v3.n3.a10}
}
Linear Extension Sums as Valuations on Cones (2012, 33 pp.)
Adrien Boussicault, Valentin Féray, Alain Lascoux, Victor Reiner
The geometric and algebraic theory of valuations on cones is ap- plied to understand identities involving summing certain rational functions over the set of linear extensions of a poset.
@article{boussicault2012linearextension,
author = {Adrien Boussicault and Valentin Féray and Alain Lascoux and Victor Reiner},
title = {{Linear Extension Sums as Valuations on Cones}},
year = {2012},
journal = {J. Algebraic Combin.},
volume = {35},
number = {4},
pages = {573--610},
doi = {10.1007/s10801-011-0316-2}
}
Noncommutative symmetric functions with matrix parameters (extended abstract) (2012, 12 pp.)
Alain Lascoux, Jean-Christophe Novelli, Jean-Yves Thibon
We define new families of noncommutative symmetric functions and quasi-symmetric functions depending on two matrices of parameters, and more generally on parameters associated with paths in a binary tree. Appropriate specializations of both matrices then give back the two-vector families of Hivert, Lascoux, and Thibon and the noncommutative Macdonald functions of Bergeron and Zabrocki. Résumé. Nous définissons de nouvelles familles de fonctions symétriques non-commutatives et de fonctions quasi-symétriques, dépendant de deux matrices de paramètres, et plus généralement, de paramètres associés à des chemins dans un arbre binaire. Pour des spécialisations appropriées, on retrouve les familles à deux vecteurs de Hivert-Lascoux-Thibon et les fonctions de Macdonald non-commutatives de Bergeron-Zabrocki.
@inproceedings{lascoux2012noncommutativesymmetric,
author = {Alain Lascoux and Jean-Christophe Novelli and Jean-Yves Thibon},
title = {{Noncommutative symmetric functions with matrix parameters (extended abstract)}},
year = {2012},
booktitle = {FPSAC 2012, Nagoya, Japan},
volume = {AR},
pages = {519--530},
series = {DMTCS proc.},
doi = {10.46298/dmtcs.3059}
}
Wrońskian of symmetric functions (2012, 9 pp.)
Alain Lascoux
We introduce a notion of Wrońskian of symmetric functions.
@article{lascoux2012wronskiansymmetric,
author = {Alain Lascoux},
title = {{Wrońskian of symmetric functions}},
year = {2012},
journal = {Wiadom. Mat.},
volume = {48},
number = {2},
pages = {135--142},
doi = {10.14708/wm.v48i2.327}
}
Logarithmic and Complex Constant Term Identities (2013, 26 pp.)
Tom Chappell, Alain Lascoux, S. Ole Warnaar, Wadim Zudilin
In recent work on the representation theory of vertex algebras re- lated to the Virasoro minimal models M (2, p), Adamović and Milas discovered logarithmic analogues of (special cases of) the famous Dyson and Morris con- stant term identities. In this paper we show how the identities of Adamović and Milas arise naturally by differentiating as-yet-conjectural complex ana- logues of the constant term identities of Dyson and Morris. We also discuss the existence of complex and logarithmic constant term identities for arbitrary root systems, and in particular prove complex and logarithmic constant term identities for the root system G2 .
@incollection{chappell2013logarithmiccomplex,
author = {Tom Chappell and Alain Lascoux and S. Ole Warnaar and Wadim Zudilin},
title = {{Logarithmic and Complex Constant Term Identities}},
year = {2013},
booktitle = {Computational and Analytical Mathematics},
volume = {50},
pages = {219--250},
publisher = {Springer, New York},
series = {Springer Proc. Math. Stat.},
eprint = {1112.3130},
archivePrefix = {arXiv},
doi = {10.1007/978-1-4614-7621-4_11}
}
Noncommutative symmetric functions with matrix parameters (2013, 21 pp.)
Alain Lascoux, Jean-Christophe Novelli, Jean-Yves Thibon
We define new families of noncommutative symmetric functions and quasi-symmetric functions depending on two matrices of parameters, and more gen- erally on parameters associated with paths in a binary tree. Appropriate specializa- tions of both matrices then give back the two-vector families of Hivert, Lascoux, and Thibon and the noncommutative Macdonald functions of Bergeron and Zabrocki.
@article{lascoux2013noncommutativesymmetric,
author = {Alain Lascoux and Jean-Christophe Novelli and Jean-Yves Thibon},
title = {{Noncommutative symmetric functions with matrix parameters}},
year = {2013},
journal = {J. Algebraic Combin.},
volume = {37},
pages = {621--642},
doi = {10.1007/s10801-012-0378-9}
}
Plaidoyer pour l'Algèbre Moderne (2013, 3 pp.)
Alain Lascoux
A brief personal opinion by Alain Lascoux advocating for an approach to modern algebra through the study of polynomials.
@unpublished{lascoux2013plaidoyerpour,
author = {Alain Lascoux},
title = {{Plaidoyer pour l'Algèbre Moderne}},
year = {2013},
note = {Unpublished}
}
Polynomial representations of the Hecke algebra of the symmetric group (2013, 18 pp.)
Alain Lascoux
We give a polynomial basis of each irreducible representation of the Hecke algebra, as well as an adjoint basis. Decompositions in these bases are obtained by mere specializations.
@article{lascoux2013polynomialrepresentations,
author = {Alain Lascoux},
title = {{Polynomial representations of the Hecke algebra of the symmetric group}},
year = {2013},
journal = {Internat. J. Algebra Comput.},
volume = {23},
number = {4},
pages = {803--818},
doi = {10.1142/s0218196713400109}
}
Polynomials (book) (2013, 354 pp.)
Alain Lascoux
@unpublished{lascoux2013polynomialsbook,
author = {Alain Lascoux},
title = {{Polynomials (book)}},
year = {2013},
note = {Unpublished}
}
An Edge-Weighted Hook Formula for Labelled Trees (2014, 19 pp.)
Valentin Féray, I. P. Goulden, Alain Lascoux
Finite sum Cauchy identity for dual Grothendieck polynomials (2014, 5 pp.)
Alain Lascoux, Hiroshi Naruse
We notice that dual Grothendieck polynomials are specializations of some vexillary Schubert polynomials. Hence they have determinantal expressions in terms of complete or elementary symmetric functions, as well as a description in terms of tableaux and Giambelli type formula. We give for them a finite sum Cauchy identity.
@article{lascoux2014finitesum,
author = {Alain Lascoux and Hiroshi Naruse},
title = {{Finite sum Cauchy identity for dual Grothendieck polynomials}},
year = {2014},
journal = {Proc. Japan Acad. Ser. A Math. Sci.},
volume = {90},
number = {7},
doi = {10.3792/pjaa.90.87}
}
Constant Term Identities and Poincaré Polynomials (2015, 27 pp.)
Gyula Károlyi, Alain Lascoux, S. Ole Warnaar
In 1982 Macdonald published his now famous constant term con- jectures for classical root systems. This paper begins with the almost trivial observation that Macdonald’s constant term identities admit an extra set of free parameters, thereby linking them to Poincaré polynomials. We then ex- ploit these extra degrees of freedom in the case of type A to give the first proof of Kadell’s orthogonality conjecture—a symmetric function generalisation of the q-Dyson conjecture or Zeilberger–Bressoud theorem. Key ingredients in our proof of Kadell’s orthogonality conjecture are multi- variable Lagrange interpolation, the scalar product for Demazure characters and (0, 1)-matrices.
@article{karolyi2015constantterm,
author = {Gyula Károlyi and Alain Lascoux and S. Ole Warnaar},
title = {{Constant Term Identities and Poincaré Polynomials}},
year = {2015},
journal = {Trans. Amer. Math. Soc.},
volume = {367},
number = {10},
pages = {6809--6836},
doi = {10.1090/tran/6119}
}