French Version  

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

1. Chern and Yang through Ice (17 pages), ps (229 KB) | dvi (68.1 KB).

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

2. Jacobian of symmetric functions (4 pages), with Piotr Pragacz, ps (103 KB) | dvi (13 KB).

   We give the Jacobian of any family of complete symmetric functions, or of power sums, in a finite number of variables.

3. Double Crystal graphs (22 pages) , ps (91.6 KB) | dvi (274 KB), (to appear in a volume dedicated to Isai Schur, published by Birkhauser).

   We show how to expand a non-symmetric Cauchy kernel 1 /∏_i+j<n (1-x_iy_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.

4. Noncommutative symmetric functions (13 pages), with Jean-Yves Thibon, Florent Hivert, ps (132 KB) | dvi (43 KB) | tex (25.2 KB).

   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.

5. Combinatorial operators on polynomials (96 pages), ps (795 KB).

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

6. The Newton interpolation formula, with more variables (6 pages), ps (108 KB) | dvi (23.1 KB).

   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.

7. About Division by 1 (7 pages), ps (133 KB) | dvi (23.8 KB).

   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.

8. Singular locus of Schubert varieties (23 pages), with Ch. Kassel & Ch. Reutenauer, ps (257 KB).

   The singular locus of a Schubert variety X_μ in the flag variety for GL_n is the union of Schubert varieties X_μ, where μ runs over a set Sg(μ) of permutations in S_n. We describe completely the maximal elements of Sg(μ) under the Bruhat order, thus determining the irreducible components of the singular locus of X_μ

9. Vertex operators and the class algebras of symmetric groups (23 pages), with Jean-Yves Thibon, ps (291 KB) | dvi (72.2 KB) | tex (42.5 KB).

   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 W_1+∞ , 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.

10. Yang-Baxter graphs, Jack and Macdonald polynomials (33 pages), ps (350 KB) | dvi (129 KB) | tex (78.3KB).

    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).

11. A filtration of the symmetric function space and a refinement of the Macdonald positivity conjecture (38 pages), with L. Lapointe and J. Morse, ps (549 KB).

    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.

12. Calculs multivariés (26 pages), colored slides, ps (320 KB).

    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

13. Transitions on Grothendieck Polynomials (15 pages), ps (205 KB) | dvi (64.9 KB) | tex (39.5 KB) | format: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

14. Young representations of the symmetric group (11 pages), ps (160 KB) | dvi (37.4 KB) | tex (24.8 KB).

    We show how to read the classical different matrices representing the symmetric group from graphs easy to generate.

15. Motzkin paths and powers of continued fractions (4 pages), ps (100 KB) | dvi (15KB) | tex (9.63 KB).

    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)

16. Sylvester's bijection between strict and odd partitions (3 pages), ps (78.3 KB) | dvi (14.1 KB) | tex (10.8 KB).

    We give a straightforward description and proof of Sylvester's bijection between strict and odd partitions

17. Orthogonal Divided Differences (26 pages), with P. Pragacz,  ps (336 KB) | dvi (104 KB) | tex (65.9 KB).

    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.

18. Une identité remarquable en théorie des partitions (16 pages), with Michel Lassalle, ps (195 KB) |  tex (27.5 KB).

    We prove an identity about plane partitions, previously conjectured in the study of shifted Jack polynomials (math.CO/9903020). The proof given is using λ-ring techniques. It would be interesting to obtain a bijective proof.

19. Ordering the Affine Symmetric Group (11 pages), ps (186 KB) | tex (58.3 KB) | dvi (36.1 KB).

    We review several descriptions of the affine symmetric group. We explicit the basis of its Bruhat order.

20. Couper les alphabets en 4 (16 pages), ps (208 KB) |  dvi (58.2 KB) | tex (36.4 KB),
    Alphabet Splitting, ps (195 KB) | dvi (53.3 KB) | tex (33.5 KB).

    We stress the importance of addition in the mathematical work of Gian-Carlo Rota (French version and EuroEnglish version).

21. Q-functions and degeneraci loci (21 pages), with P. Pragacz, ps (295 KB)  | tex (AMSTeX, 59.9 KB). 

    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.

22. Factorization of Kazhdan-Lusztig elements for Grassmannians (12 pages), with A. Kirillov, ps (364 KB).

    We show that the Kazhdan-Lusztig 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.

23. Square-Ice enumeration (15 pages), ps (411 KB) | dvi (67.3 KB).

    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.

24. Operator Calculus for Q-polynomials and Schubert polynomials (38 pages), with P. Pragacz, Advances in Math. 140(1998)1-43, ps (446 KB) | dvi (166 KB).

    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.

25. Determinantal Expressions for Macdonald polynomials (21 pages), with L.Lapointe and J.Morse; Intern.Math.Res.Not.(1998)957-978, ps (267 KB) | dvi (89.8 KB).

    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.

26. Caractéristique d'Euler-Poincaré selon Hirzebruch (12 pages), ps (214 KB) | dvi (54.8 KB).

    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.

27. The Plactic Monoid (29 pages), with B. Leclerc & J-Y. Thibon, preliminary draft of a chapter for the new Lothaire book "Algebraic Combinatorics on Words" , ps (351 KB).

    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.

28. Factorization in Schubert cells (30 pages), with Ch. Kassel & Ch. Reutenauer, pdf (255 KB) |dvi (126 KB, 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.

29. Ordre de Bruhat sur le groupe symétrique (10 pages), ps (373 KB).

    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.

30. Notes on Interpolation in one and several variables (40 pages), ps (368 KB) | dvi (144 KB).

    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 exercises (in French) are available on request.

31. Young's natural idempotents as polynomials (9 pages), ps (145 KB) | dvi (32.6 KB).

    Coding permutations as monomials, one obtains a compact expression of representatives of Young's natural idempotents for the symmetric group, or the Hecke algebra.

32. Potentiel Yin sur le groupe symétrique (12 pages), ps (166 KB).

33. Pour le Monoïde Plaxique (M.P. Schützenberger) (7 pages), ps (95 KB) | dvi (19.4 KB).

34. Treillis et bases des groupes de Coxeter (includes polynômes de Kazhdan-Lusztig pour les variétés de Schubert vexillaires), (38 pages), with M.P. Schützenberger, ps (369 KB) | dvi (146 KB).

35. Opérateurs différentiels sur l'anneau des polynômes symétriques (30 pages), (Manuscrit 1991), ps (415 KB).

French Version