## Recent Publications

For articles written with members of the Phalanstère, go to the list of publications of Jean-Yves Thibon.

1. Plaidoyer pour l'algèbre moderne

2. Polynomes

3. 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 so-called q-Dyson conjecture.

4. 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.

5. How many alphabets can a Schur function accomodate ? 4.

6. 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 as-yet-conjectural complex analogues of the constant term identities of Dyson and Morris.

7. Noncommutative symmetric functions with matrix parameters, with 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.

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

9. Hankel Pfaffians, Discriminants and Kazhdan-Lusztig bases. We use Kazhdan-Lusztig bases of representations of the symmetric group to express Pfaffians with entries $(a_i-a_j) h_{i+j}$. 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 q-discriminant.

10. 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.

11. 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.

12. Deformed Kazhdan-Lusztig elements and Macdonald polynomials, with Jan De Gier et Mark Sorrell. We introduce deformations of Kazhdan-Lusztig 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.

13. Generalisation of Scott permanent identity; Scott considered the determinant of 1/(y-z)^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^n-a and z^n-b. We give the same specialisation for the determinant 1/\prod_x( xy-z) , where {x} is an arbitrary set of indeterminates. The case of the Gaudin-Izergin-Korepin determinant is for {x}={q,1/q}.

14. Branching rules for symmetric Macdonald polynomials , with S.Ole Warnaar. A one-parameter 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 q-difference equation. We also prove a new multiple q-Gauss summation formula and several further results for sl_n basic hypergeometric series.

15. 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.

16. Nonsymmetric interpolation Macdonald polynomials and g_n basic hypergeometric series with 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 $\gn$. Our main results include a new $q$-binomial theorem, new $q$-Gauss sum, and several transformation formulae for $\gn$ series.

17. Schubert and Macdonald for Dummies; Slides; Schubert and (non-symmetric) 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.

18. Gaudin functions, and Euler-Poincaré 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_i-y_j)(x_i-ty_j) ... (x_i-t^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 x-variables into the y-variables, 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 Euler-Poincar� characteristics of the flag manifold. As a corollary, we recover a result of Warnaar about the generating function of Macdonald polynomials.

19. Adding \pm 1 to the argument of a Hall-Littlewood 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 Hall-Littlewood 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 Rogers-Ramanujan identities.

20. Non symmetric Cauchy kernels for the classical Groups with Amy Fu. We give non-symmetric 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.

21. 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 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. 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/

22. 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_i-x_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.

23. 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.

24. Non-Symmetric Hall-Littlewood Polynomials with F. Descouens, (pdf, 15 p.), FPSAC 2005, dedicated to Adriano Garsia, 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.

25. 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 Geck-Rouquier, and to the basis of Jones, using matrices of change of bases of the ring of symmetric polynomials.

26. Differential Equation of a Plane Curve(pdf), Bull. Sc. Math. 130 (2006) 354-359 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.

27. Some composition determinants (pdf, 9 p.) with J.M. Brunat, C. Krattenthaler, A. Montes, Linear Alg. Appl. 416 (2006) 355-364. 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), 459--471] and ["A polynomial generalization of the power-compositions determinant," Linear Multilinear Algebra (to appear)], and they prove two conjectures of the second author

28. 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 Yang-Baxter bases of irreducible representations occur naturally from this divided difference approach.

29. Addition of 1, (Séminaire Lotharingien, March 04; 8 pages ps.gz) We show that some classical identities about multiplicative functions, and the Riemann zeta-function, may conveniently be interpreted in terms of the addition or substraction of 1 to some alphabets.

30. Bezoutiants and Orthogonal Polynomials, with Piotr Pragacz(14 pages . ps.gz) We interpret the Bezoutiant of two univariate polynomials as a Christoffel-Darboux kernel. This gives orthogonality properties of remainders and reproducing properties of the Bezoutiant.

31. 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.

32. $q$-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.

33. Evaluation of Some Hankel Determinants , with Qing-Hu Hou and Yang-Ping 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 q-integers (Adv. Appl. M., volume Robbins).

34. 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).

35. 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 Zhi-Guo Liu. Some classical $q$-series identities and bibasic identities are a consequence of it.

36. 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 non-commutative Schubert polynomials are obtained by reading the columns of a two-dimensional Cauchy kernel.

37. ContinuedFractionsForRogersSzego, with Qing-Hu Hou and Yang-Ping Mu (11 pages ps.gz) or ConFracRogersSzego.dvi

We evaluate different Hankel determinants of Rogers-Szego polynomials, and deduce continued fraction expansions.

38. q-Identities from Lagrange and Newton,, with Amy Fu (pdf, 6 p.) Combining Newton and Lagrange interpolation, we give q-identities which generalize results of Van Hamme, Uchimura, Dilcher and Prodinger.

39. 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.

40. Schubert polynomials, (50 slides ps.gz in French)

Three elementary self-contained ways of obtaining Schubert polynomials: interpolation, diagonalization of a Cauchy kernel, or coefficients of Yang-Baxter elements.

41. Littlewood's formulas, (10 pages ps.gz) or Littlewood.dvi.gz

Litllewood gave expansions of products of the type $\prod 1/(1-ab)$. We show that the method of Littlewood covers several generalizations recently published.

42. Chern and Yang through Ice, (17 pages ps.gz) or ChernYang.dvi.gz

Characteristic classes for flags of vector bundles, Yang-Baxter coefficients and Grothendieck polynomials can be expressed by a simple statistics on alternating-sign matrices.

43. 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.

44. Double Crystal graphs, (22 pages .ps.gz) or VolSchur.dvi (in a volume dedicated to Schur, Progress in Math 210, Birkhauser (2003)95-114).

We show how to expand a non-symmetric Cauchy kernel 1 /\prod_{i+j< n} (1-x_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.

45. Noncommutative symmetric functions., with Jean-Yves Thibon, Florent Hivert(13 pages ps.gz) or NonCommSym.dvi.gz or NonCommSym.tex

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.

46. Combinatorial operators on polynomials (96 pages ps.gz)

Notes of a course on Symmetric functions and Combinatorial Operators on polynomials'', North-Carolina June 01.

47. 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.

48. 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.

49. 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}

50. Vertex operators and the class algebras of symmetric groups ., with Jean-Yves Thibon, (23 pages ps.gz) or VertexOpAndCenter.dvi.gz or VertexOpAndCenter.tex

We exhibit a vertex operator which implements multiplication by power-sums of Jucys-Murphy 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.

51. Yang-Baxter 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 Yang-Baxter relations (in conjunction with a graph associated to the extended affine symmetric group).

52. 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.

53. 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.univ-mlv.fr/~ace

54. Transitions on Grothendieck Polynomials. (15 pages ps.gz) or TransOnGroth.dvi.gz or TransOnGroth.tex ws-p8-50x6-00.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 ws-p8-50x6-00.tex

55. 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.

56. 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)

57. 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

58. 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.

59. 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.

60. 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.

61. 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 Gian-Carlo Rota (French version and EuroEnglish version).

62. Q-functions 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 Q-functions) for the fundamental classes of degeneracy loci associated with vector bundle maps given locally by (rectangular) matrices which are symmetric or antisymmetric.

63. Factorization of Kazhdan-Lusztig elements for Grassmannians with A. Kirillov. 12 pages (80 Ko ps.gz).

We show that the Kazhdan-Lusztig basis elements Cw 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 (Ti+fj(v)), where fj are rational functions.

64. Square-Ice enumeration 15 pages (89 Ko, ps.gz or 21 Ko, dvi.gz SquareIce.dvi.gz).

Enumeration of square-ice models, or alternating-sign matrices lead to the study of Cauchy type determinants of entries (1/(x-y)(qx-y)) or (1/(x-y)(x-y+ 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.

65. Operator Calculus for Q-polynomials and Schubert polynomials with P. Pragacz, Advances in Math. 140(1998)1-43, 38 pages (167 Ko, ps.gz or 59 Ko, dvi.gz OperatorCalculus.dvi.gz).

We choose products of Schubert polynomials and Q-functions 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.

66. Determinantal Expressions for Macdonald polynomials (with L.Lapointe and J.Morse; Intern.Math.Res.Not.(1998)957-978) 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 lambda-rings. We obtain a determinantal expression of Macdonald polynomials.

67. Caractéristique d'Euler-Poincaré selon Hirzebruch 12 pages, (85 Ko, ps.gz ou 23 Ko, dvi.gz Hirzebruch.dvi.gz).

We indicate how Hecke algebras, Yang-Baxter equation, Hall-Littlewood polynomials, Macdonald polynomials, can be traced back to the parameter $y$ that Hirzebruch introduced in his study of Riemann-Roch theorem.

68. The Plactic Monoid avec B. Leclerc & J-Y. 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.

69. 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.

70. 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 Kazdhan-Lusztig basis of the Hecke algebra of the symmetric group: vanishing or not of Kazdhan-Lusztig polynomials ensure that permutations are comparable or not. We explicit these polynomials in the case of vexillary permutations.

71. 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 self-contained 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.

72. 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.

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

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

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

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