 Plaidoyer pour l'algèbre moderne
 Polynomes

Constant term identities and Poincaré polynomials,
with Gyula Károlyi and S.Ole Warnaar.
We remark that Macdonald's
constant term identities admit an extra set of parameters.
We use this in type A to prove Kadell's orthogonality
conjecture  a symmetric function generalisation of the socalled qDyson
conjecture.

Polynomial representations of the Hecke algebra
of the symmetric group.
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.

How many alphabets can a Schur function accomodate ?
4.

Logarithmic and complex constant term identities,
with Tom Chappell, S. Ole Warnaar, Wadim Zudilin.
Adamovic and Milas discovered logarithmic analogues of (special cases
of) the famous Dyson and Morris constant term identities.
We show how the identities
of Adamovic and Milas arise naturally by
differentiating asyetconjectural complex analogues of
the constant term identities of Dyson and Morris.

Noncommutative symmetric functions with matrix parameters,
with JeanChristophe Novelli, JeanYves Thibon.
We define new families of noncommutative symmetric functions and
quasisymmetric 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 twovector
families of Hivert, Lascoux, and Thibon
and the noncommutative Macdonald functions of Bergeron and Zabrocki.

Wronskian of symmetric functions.
We introduce a notion of Wronskian of symmetric functions.
(Wiadomosci Matematyczne, Tom 48, Nr 2 (2012) )

Hankel Pfaffians, Discriminants and KazhdanLusztig bases.
We use KazhdanLusztig bases of representations of the
symmetric group to express Pfaffians with entries
$(a_ia_j) h_{i+j}$. In the case where the parameters a_i
are specialized to successive powers of q, and the h_i
are complete functions, we obtain the qdiscriminant.

Idempotents with polynomial coefficients
We combine Young idempotents in the group algebra of the symmetric group
with the action of the symmetric group on products of Vandermonde
determinants to obtain idempotents with polynomial coefficients.

Linear extension sums as valuations of cones,
with Adrien Boussicault, Valentin Feray, Victor Reiner.
The geometric and algebraic theory of valuations on cones is
applied to understand identities involving summing certain
rational functions over the set of linear extensions of a poset.

Deformed KazhdanLusztig elements and Macdonald polynomials,
with Jan De Gier et Mark Sorrell.
We introduce deformations of KazhdanLusztig elements
and degenerate nonsymmetric Macdonald
polynomials, both of which form a basis of a
polynomial representation of the Hecke algebra.
We give explicit integral formulas and
transition matrices for these polynomials.

Generalisation of Scott permanent identity;
Scott considered the determinant of 1/(yz)^2, with
y,z running over two sets Y,Z of size n,
and determined its specialisation when Y and Z are the roots of
y^na and z^nb. We give the same specialisation for
the determinant 1/\prod_x( xyz) , where {x} is an arbitrary set
of indeterminates. The case of the GaudinIzerginKorepin determinant
is for {x}={q,1/q}.

Branching rules for symmetric Macdonald polynomials ,
with S.Ole Warnaar.
A oneparameter generalisation of the symmetric Macdonald polynomials
and interpolations Macdonald polynomials is studied from the point of view
of branching rules. We establish a Pieri formula, evaluation symmetry,
principal specialisation formula and qdifference equation.
We also prove a new multiple qGauss summation formula and several
further results for sl_n basic hypergeometric series.

Thom polynomials and Schur functions: the singularities A_3
with P.Pragacz.
Combining the ``method of restriction equations'' of
Rimanyi et al. with the techniques
of symmetric functions, we establish the Schur function expansions
of the Thom polynomials for the Morin singularities
A_3.

Nonsymmetric interpolation Macdonald polynomials and g_n basic hypergeometric series
with Eric M. Rains, S. Ole Warnaar.
The KnopSahi interpolation Macdonald polynomials are inhomogeneous and
nonsymmetric generalisations of the wellknown Macdonald polynomials.
In this paper we apply the interpolation Macdonald polynomials to study
a new type of basic hypergeometric series of type $\gn$.
Our main results include a new $q$binomial theorem, new $q$Gauss sum, and
several transformation formulae for $\gn$ series.

Schubert and Macdonald for Dummies;
Slides;
Schubert and (nonsymmetric) Macdonald polynomials
are two linear bases of the ring of polynomials which can be
characterized by vanishing conditions.
We show that both families satisfy similar branching rules
related to the multiplication by a single variable.
These rules are sufficient to recover a great part of the theory
of Schubert and Macdonald polynomials.

Gaudin functions, and EulerPoincaré characteristics;
Slides with figures;
Given two positive integers n,r, we define the Gaudin function
of level r to be quotient of the numerator of the determinant
det(1/ ((x_iy_j)(x_ity_j) ... (x_it^r y_j)), i,j=1..n,
by the two Vandermonde in x and y.
We show that it can be characterized by specializing the xvariables
into the yvariables, multiplied by powers of t.
This allows us to obtain the Gaudin function of level 1 (due to Korepin and
Izergin) as the image of a resultant under the
the EulerPoincar� characteristics of the flag manifold.
As a corollary, we recover a result of Warnaar about the generating function
of Macdonald polynomials.

Adding \pm 1 to the argument of a HallLittlewood polynomial
(Séminaire Lotharingien, vol 54).
Shifting by 1 powers sums: p_i > p_i +1 induces a transformation
on symmetric functions that we detail in the case of HallLittlewood
polynomials. By iteration, this gives a description of these
polynomials in terms of plane partitions, as well as some
generating functions. We recover in particular an identity
of Warnaar related to RogersRamanujan identities.
 Non symmetric Cauchy kernels for the classical Groups
with Amy Fu.
We give nonsymmetric versions of the
Cauchy kernel and Littlewood's kernels, corresponding to the types
A,B,C,D of the classical groups.
We show that these new kernels are diagonal in the basis of two families of key
polynomials obtained as images of dominant monomials under
divided differences. We define new scalar products
such that the two families of key polynomials are adjoint to each other.

The 6 Vertex Model and Schubert Polynomials
Journal SIGMA 3 (2007), 029, 12 pages.
We enumerate staircases with fixed left and right columns.
These objects correspond to
iceconfigurations, 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.
Contribution to the Vadim Kuznetsov Memorial Issue on
Integrable Systems and Related Topics, published in SIGMA (Symmetry,
Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA/
 Pfaffians and Representations of
the Symmetric Group (pdf, 28 p.)
Pfaffians of matrices with entries z[i,j]/(x_i+x_j), or determinants
of matrices with entries z[i,j]/(x_ix_j), where the antisymmetrical
indeterminates z[i,j] satisfy the Plucker 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.
 Muir and Littlewood's Products,
with Amy Fu, (pdf, 9 p.), to appear in Linear Alg.,
We answer a question of Muir, relating it
to different determinantal expressions for the products
\prod(y x_ix_j), and for the products of these functions by an
arbitrary Schur function.
 NonSymmetric HallLittlewood Polynomials
with F. Descouens, (pdf, 15 p.), FPSAC 2005, dedicated to Adriano Garsia,
Using the action of the YangBaxter elements of the Hecke algebra on
polynomials, we define two bases of polynomials in n variables.
The HallLittlewood 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.
 Center of the Hecke Algebra and Symmetric Functions (pdf, 25 p.) submitted to J.Alg.
We give half a dozen bases of the Hecke algebra of the symmetric
group, and relate them to the basis of GeckRouquier, and to the basis
of Jones, using matrices of change of bases of the ring of
symmetric polynomials.
 Differential Equation of a Plane Curve(pdf), Bull. Sc. Math. 130 (2006) 354359
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.
 Some composition determinants
(pdf, 9 p.) with J.M. Brunat, C. Krattenthaler, A. Montes,
Linear Alg. Appl. 416 (2006) 355364.
We compute two parametric determinants in which rows and columns are indexed by compositions, where in one determinant the entries are products of binomial coefficients, while in the other the entries are products of powers. These results generalize previous determinant evaluations due to the first and third author [SIAM J. Matrix Anal. Appl. 23 (2001), 459471] and ["A polynomial generalization of the powercompositions determinant," Linear Multilinear Algebra (to appear)], and they prove two conjectures of the second author
 Operators on Polynomials (ps),
or Operators on Polynomials (dvi),
ACE Summer School, July 2004, 70 pages
The symmetric group acts in different ways on the ring of polynomials.
Instead of only permuting indeterminates, one can follow Newton
and use divided differences, and even deform them and
obtain an Hecke algebra of operators.
We shall show how the representation theory of the symmetric
group, the coinvariant ring of the symmetric group (cohomoly ring
of the flag manifold), the ring
of polynomials as a free module over symmetric ones and
YangBaxter bases of irreducible representations occur naturally
from this divided difference approach.
 Addition of 1,
(Séminaire Lotharingien, March 04; 8 pages ps.gz)
We show that some classical identities about multiplicative functions,
and the Riemann zetafunction, may conveniently be interpreted
in terms of the addition or substraction of 1 to some alphabets.
 Bezoutiants and Orthogonal Polynomials,
with Piotr Pragacz(14 pages . ps.gz)
We interpret the Bezoutiant of two univariate polynomials as
a ChristoffelDarboux kernel.
This gives orthogonality properties of remainders
and reproducing properties of the Bezoutiant.

The Differential Equation of a Plane Curve
(7 pages, ps.gz, talk March 04, Tianjin )
y"=0 is the equation of a line in the Cartesian plane. Monge gave
the differential equation satisfied by conics. More generally, the
equations of planar curves are projective invariants that one can write
using symmetric functions.

$q$Identities related to overpartitions and divisor functions
with Amy Fu (pdf, 6 p.)
We generalize and prove conjectures of
Corteel and Lovejoy, related to overpartitions and divisor functions.

Evaluation of Some Hankel Determinants
, with QingHu Hou and YangPing Mu (9 pages ps.gz)
We evaluate Hankel determinants
of Meixner polynomials
associated to the series
exp (\sum a [i] z^i/i), where [1],[2]...
are the qintegers (Adv. Appl. M., volume Robbins).

Partition Analysis and Symmetrizing Operators,
with Amy Fu (pdf, 5 p.)
Using a symmetrizing operator,
we give a new expression for the Omega operator used by MacMahon
in Partition Analysis, and given a new life by Andrews
(JCTA 2004).
 Rational Interpolation and
Basic Hypergeometric Series, with Amy Fu (pdf, 11 p.)
We give a Newton type rational interpolation formula.
It contains as a special case the original Newton interpolation, as
well as the recent interpolation formula of ZhiGuo Liu.
Some classical $q$series identities and bibasic identities are a
consequence of it.
 Schubert and Grothendieck,
(Séminaire Lotharingien, March 03; ps.gz; 23 p. in French +
6 pages summary in English)
or Schubert&Grothendieck Slides
(slides  text different)
We give a dozen formulas concerning Schubert and Grothendieck polynomials,
and their interrelations, half of them being new, and most of them interesting.
In particular, we explicit the decomposition of Schubert polynomials
as positive sums of Grothendieck polynomials, and show that noncommutative
Schubert polynomials are obtained by reading the columns of a twodimensional
Cauchy kernel.
 ContinuedFractionsForRogersSzego,
with QingHu Hou and YangPing Mu
(11 pages ps.gz) or
ConFracRogersSzego.dvi
We evaluate different Hankel determinants of RogersSzego
polynomials, and deduce continued fraction expansions.

qIdentities from Lagrange and Newton,,
with Amy Fu (pdf, 6 p.)
Combining Newton and Lagrange
interpolation, we give qidentities which generalize results of
Van Hamme, Uchimura, Dilcher and Prodinger.
 SylvesterSumsForRemainders,
with Piotr Pragacz(20 pages . ps.gz) or
SylvesterSumsForRemainders.dvi
We comment and prove the formulas that Sylvester gave about
the succesive remainders of two polynomials in one variable,
in terms of the roots of the two polynomials.
 Schubert polynomials,
(50 slides ps.gz in French)
Three elementary selfcontained ways of obtaining Schubert polynomials:
interpolation, diagonalization of a Cauchy kernel, or coefficients
of YangBaxter elements.
 Littlewood's formulas,
(10 pages ps.gz)
or Littlewood.dvi.gz
Litllewood gave expansions of products of the type
$\prod 1/(1ab)$. We show that the method of Littlewood covers several
generalizations recently published.
 Chern and Yang through Ice,
(17 pages ps.gz)
or ChernYang.dvi.gz
Characteristic classes for flags of vector bundles, YangBaxter coefficients
and Grothendieck polynomials can be expressed by a simple statistics on
alternatingsign matrices.
 Jacobian of symmetric functions.
(4 pages ps.gz)
or Jacobien.dvi.gz
with Piotr Pragacz.
We give the Jacobian of any family of complete symmetric functions,
or of power sums, in a finite number of variables.
 Double Crystal graphs,
(22 pages .ps.gz) or
VolSchur.dvi
(in a volume dedicated to Schur, Progress in Math 210, Birkhauser
(2003)95114).
We show how to expand a nonsymmetric Cauchy kernel
1 /\prod_{i+j< n} (1x_i y_j) in the basis of
Demazure characters. The construction involves
using the left and right structure of crystal graphs on words,
and mostly reduces to properties of the jeu de taquin.
 Noncommutative
symmetric functions., with JeanYves Thibon,
Florent Hivert(13 pages ps.gz) or
NonCommSym.dvi.gz or
NonCommSym.tex
We define twoparameter families of noncommutative symmetric functions and
quasisymmetric functions, which appear to be the proper analogues of the
Macdonald symmetric functions in these settings.
 Combinatorial operators
on polynomials (96 pages ps.gz)
Notes of a course on ``Symmetric functions and Combinatorial Operators
on polynomials'', NorthCarolina June 01.
 The Newton interpolation
formula, with more variables (6 pages ps.gz)
or NewtonInterp.dvi.gz
We give the generalization of Newton's interpolation formula
to several variables, assuming no previous knowledge
of Schubert polynomials, but using only simple vanishing conditions.
This is an old work with M.P. Schutzenberger.
 About Division by 1
(7 pages ps.gz) or
DivBy1.dvi.gz
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.
 Singular locus of Schubert varieties
with Ch. Kassel &
Ch. Reutenauer. (23 pages ps.gz)
The singular locus of a Schubert variety X_{\mu} in the flag
variety
for GL_n is the union of Schubert varieties X_{\nu}, where \nu runs over a set
Sg(\mu) of permutations in S_n. We describe completely the maximal elements of
Sg(\mu) under the Bruhat order, thus determining the irreducible components of
the singular locus of X_{\mu}
 Vertex operators and the class algebras of
symmetric groups ., with JeanYves Thibon, (23 pages ps.gz) or
VertexOpAndCenter.dvi.gz or
VertexOpAndCenter.tex
We exhibit a vertex operator which implements multiplication by
powersums of JucysMurphy elements in the centers of the group algebras
of all symmetric groups simultaneously. The coefficients of this operator
generate a representation of ${\cal W}_{1+\infty}$, to which operators
multiplying by normalized conjugacy classes are also shown to belong.
A new derivation
of such operators based on matrix integrals is proposed, and our vertex
operator is used to give an alternative approach to the polynomial
functions on Young diagrams introduced by Kerov and Olshanski.
 YangBaxter graphs, Jack and
Macdonald polynomials. (33 pages ps.gz) or
YangRota.dvi.gz or
YangRota.tex.gz
The different varieties of Jack and Macdonald polynomials
can be computed using YangBaxter relations (in conjunction
with a graph associated to the extended affine symmetric group).

A filtration of the symmetric function space and a refinement of the Macdonald
positivity conjecture
(with L.Lapointe and J.Morse 38 pages)
For each integer $k$, we introduce a new family of symmetric polynomials,
constructed from sums of tableaux using the charge statistic. We conjecture
that the Macdonald polynomials indexed by partitions bounded by $k$ expand
positively in terms of these polynomials.
 Calculs multivariés. .
(26 pages .ps, colored slides)
We describe tools related to the symmetric group to compute
functions of several variables. Many of them have been implemented
as a Maple library (ACE) and can be found on the site
http://phalanstere.univmlv.fr/~ace
 Transitions on Grothendieck
Polynomials. (15 pages ps.gz) or
TransOnGroth.dvi.gz or
TransOnGroth.tex
wsp850x600.cls
We describe how general Grothendieck Polynomials (representatives of Schubert
varieties in the Grothendieck ring of flag manifolds)
are related to those for Grasmann manifolds, which themselves are
deformations of Schur functions.
Compile with : latex wsp850x600.tex
 Young representations of the
symmetric group. ( 11pages ps.gz) or
YoungRep.dvi.gz or
YoungRep.tex
We show how to read the classical different matrices representing the
symmetric group from graphs easy to generate.
 Motzkin paths and powers of continued
fractions. ( 4pages ps.gz) or
Motzkin.dvi.gz or
Motzkin.tex
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.
(submitted to the Seminaire Lotharingien de Combinatoire)

Sylvester's bijection between strict and odd partitions (3pages )
ps.gz)
SylvesterBij.dvi.gz or
SylvesterBij.tex
We give a straightforward description and proof of Sylvester's bijection
between strict and odd partitions

Orthogonal Divided Differences
with P. Pragacz,
26 pages (128 Ko, ps.gz) or
(40 Ko, dvi.gz)
OrthDivDiff.dvi.gz or
OrthDivDiff.tex
Using orthogonal divided differences, we study the ring of polynomial
as a free module over the invariants of Weyl groups of type D.
We apply this description to the cohomology ring of the corresponding
flag variety.
 Une identité
remarquable en théorie des partitions (ps.gz)
avec Michel Lassalle,
16 pages, 77 Ko,
IdentitePartitions.tex.gz) 9Ko.
We prove an identity about plane partitions, previously conjectured in the
study of shifted Jack polynomials (math.CO/9903020). The proof given is
using $\lambda$ring techniques. It would be interesting to obtain a
bijective proof.
 Ordering the Affine Symmetric
Group(ps)
(11 pages, 63 Ko, ps.gz) or
Ordering the Affine Symmetric
Group(Plain tex, 29Ko)) or
Ordering the Affine Symmetric
Group(dvi.gz, 17Ko))
We review several descriptions of the affine symmetric group.
We explicit the basis of its Bruhat order.
 Couper les alphabets en 4(ps)
16 pages (80 Ko, ps.gz or
26 Ko, dvi.gz or 37 Ko, tex
Couper les alphabets en 4(tex))
Couper les alphabets en 4(dvi)).
or Alphabet Splitting(ps)
or Alphabet Splitting(dvi)
or Alphabet Splitting(tex)
We stress the importance of addition in the mathematical work
of GianCarlo Rota (French version and EuroEnglish version).
 Qfunctions and
degeneraci loci
with P. Pragacz,
21 pages (116 Ko, ps.gz ou
20 Ko, AMSTeX
Qfunctions_and_degeneraci.tex.gz).
We give formulas (involving Schur Qfunctions)
for the fundamental classes of degeneracy loci associated
with vector bundle maps given locally by (rectangular) matrices
which are symmetric or antisymmetric.
 Factorization of
KazhdanLusztig elements for Grassmannians
with A. Kirillov.
12 pages (80 Ko ps.gz).
We show that the KazhdanLusztig basis elements C_{w} of the Hecke
algebra of the symmetric group, when w corresponds to a
Schubert subvariety of a Grassmann variety, can be
written as a product of factors of the form (T_{i}+f_{j}(v)),
where f_{j}
are rational functions.
 SquareIce enumeration
15 pages (89 Ko, ps.gz or
21 Ko, dvi.gz
SquareIce.dvi.gz).
Enumeration of squareice models, or alternatingsign matrices lead to the
study of Cauchy type determinants of entries (1/(xy)(qxy)) or
(1/(xy)(xy+ g)), where {x} and{y} are two sets of the same cardinal, and
q,g are constants. Up to trivial factors, these determinants are symmetric
functions in {x} and {y} that we show how to explicit by factorizing them.

Operator Calculus for Qpolynomials and Schubert polynomials
with P. Pragacz, Advances in Math. 140(1998)143,
38 pages (167 Ko, ps.gz or
59 Ko, dvi.gz
OperatorCalculus.dvi.gz).
We choose products of Schubert polynomials and Qfunctions as a basis
of the ring of polynomials as a free module over symmetric polynomials
in the squares of the variables. We show in particular how to express vertex
operators for P and Q Schur functions in terms of divided differences for the
hyperoctahedral group. This gives a description of the cohomology ring of a
Lagrangian flag manifold.

Determinantal Expressions for Macdonald polynomials
(with L.Lapointe and J.Morse; Intern.Math.Res.Not.(1998)957978)
21 pages (99 Ko, ps.gz or
32 Ko, dvi.gz
Mac_Det.dvi.gz).
We show how to express Macdonald operators and creation operators in
terms of divided differences and lambdarings. We obtain a determinantal
expression of Macdonald polynomials.

Caractéristique d'EulerPoincaré selon Hirzebruch
12 pages, (85 Ko, ps.gz ou
23 Ko, dvi.gz
Hirzebruch.dvi.gz).
We indicate how Hecke algebras, YangBaxter equation, HallLittlewood
polynomials, Macdonald polynomials, can be traced back to the parameter $y$
that Hirzebruch introduced in his study of RiemannRoch theorem.
 The Plactic Monoid
avec B. Leclerc &
JY. Thibon,
preliminary draft of a chapter for
the new Lothaire book "Algebraic Combinatorics on Words"
29 pages, (122 Ko, ps.gz
plactic.ps.gz.gz).
This survey article reviews the structure of monoid of the set of
Young tableaux, its poset struture, and the lifting of symmetric functions
to the level of the free algebra. This point of view has been developped
together with M.P. Schützenberger.
 Factorization in
Schubert cells
with Ch. Kassel &
Ch. Reutenauer.
30 pages (260 Ko, pdf ou
49 Ko, dvi.gz
Factorization_Schubert.dvi.gz, without figures).
Let P_i(x) be a matrix, obtained from the standard matrix representing the
simple transposition (i,i+1) by adding a parameter x in position (i,i).
Then reduced products of such matrices parametrize Schubert varieties.
Change of parametrizations are polynomial. Moreover, one recovers
many classical combinatorial objects (Rothe diagrams, balanced tableaux, ...)
from such matrices.
 Ordre de Bruhat sur le
groupe symétrique.
10 pages (87 Ko, ps.gz).
One usually defines the Bruhat order on the symmetric group by subwords of
reduced decompositions or by comparison of tableaux (Ehresmann).
M.P. Schutzenberger and I prefered to embedd the symmetric group into a
distributive lattice. The most powerful method, however, is to use the
KazdhanLusztig basis of the Hecke algebra of the symmetric group:
vanishing or not of KazdhanLusztig polynomials ensure that permutations are
comparable or not. We explicit these polynomials in the case of vexillary
permutations.

Notes on Interpolation in one and several variables,
40 pages (137Ko, ps.gz, ou
54Ko, dvi.gz
interp.dvi.gz).
Divided differences are a powerful tool on functions of several
variables. We show how to recover from them the classical interpolation
formulas, as well as the multivariate extension of Newton's interpolation.
We give a compact and selfcontained exposition of Schubert polynomials,
as a basis of the free module of polynomials over the ring of symmetric
polynomials. Many exercices (in French) are available on request.

Young's natural idempotents as polynomials,
9 pages (59Ko, ps.gz ou
14Ko, fichier dvi.gz
YoungIdemp2Pol.dvi.gz).
Coding permutations as monomials, one obtains a compact expression of
representatives of Young's natural idempotents for the symmetric group,
or the Hecke algebra.

Potentiel Yin sur le groupe symétrique,
12 pages (59Ko, ps.gz).

Pour le Monoïde Plaxique (M.P. Schützenberger),
7 pages (40Ko, ps.gz, ou fichier dvi.gz
plaxique.dvi.gz).

Treillis et bases des groupes de Coxeter,
(includes polynômes de KazhdanLusztig pour les variétés de
Schubert vexillaires), 38 pages (138Ko, ps.gz),
(with M.P. Schützenberger),
(60Ko, dvi.gz
treillisBruhat95.dvi.gz).

Opérateurs différentiels sur l'anneau des polynômes
symétriques, (Manuscrit 1991),
30 pages (160Ko, ps.gz).