Alain Lascoux

Université Gustave Eiffel · Institut d'électronique et d’'informatique Gaspard Monge (IGM)

In 1982 Macdonald published his now famous constant term conjectures 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 Poincare polynomials. We then exploit these extra degrees of freedom in the case of type A to...

In recent work on the representation theory of vertex algebras related to the Virasoro minimal models M(2,p), Adamovic and Milas discovered logarithmic analogues of (special cases of) the famous Dyson and Morris constant term identities. In this paper we show how the identities of Adamovic and Milas arise naturally by differentiating as-yet-conject...

International audience 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...

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 polynomial...

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 Macdona...

A one-parameter generalisation R_{\lambda}(X;b) 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 for R_{\lambda}(X;b). We also prove a new multiple q-Gauss...

A one-parameter rational function generalisation Rλ(X;b)Rλ(X;b) of the symmetric Macdonald polynomial and interpolation Macdonald polynomial is studied from the point of view of branching rules. We establish a Pieri formula, evaluation symmetry, principal specialisation formula and q-difference equation for Rλ(X;b)Rλ(X;b). Our main motivation for s...

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 singularities A 3 : (C, 0) → (C+k, 0) for any nonnegative integer k. © 2010 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved...

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 gl_n. Our main results include a new q-binomial theorem, a new q-Gauss sum, and severa...

Combining the "method of restriction equations" of Rim\'anyi et al. with the techniques of symmetric functions, we establish the Schur function expansions of the Thom polynomials for the Morin singularities $A_3: ({\bf C}^{\bullet},0)\to ({\bf C}^{\bullet + k},0)$ for any nonnegative integer $k$.

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 t...

Using divided differences 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 differences on some distinguished symmetric polynomials ( e P - polynomials) and relate it to vertex operators....

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 difference...

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

We give several new formulas which are useful for Schubert Calculus associated with the orthogonal groups and related orthogonal degeneracy loci.

We give closed-form formulas for the fundamental classes of degeneracy loci associated with vector bundle maps given locally by (not necessary square) matrices which are symmetric (respectively skew-symmetric) with respect to the main diagonal. Our description uses essentially Schur Q-polynomials of a bundle and is based on a push-forward formula f...

that maps birationally onto D and for which one can compute its class [Z]. Usually this is because [Z] is the zero locus of a section of some bundle whose rank is equal to codimX 0 Z so the class [Z] is evaluated to be the top Chern class of the bundle. For example, this pattern was used in [J-L-P] and many other papers (see [F]). To compute the fu...

this paper. 2 Background 2.1 Outline of the commutative theory Here is a brief review of the classical theory of symmetric functions. A standard reference is Macdonald's book [Mcd]. The notations used here are those of [LS1]. Denote by X = fx 1 ; x 2 ; : : : g an infinite set of commutative indeterminates, which will be called a (commutative) alpha...

Classes dual to Schubert cycles constitute a basis on the cohomology ring of the flag manifold F , self-adjoint up to indexation with respect to the intersection form. Here, we study the bilinear form (X ; Y ) := hX Delta Y ; c(F)i where X;Y are cocycles, c(F) is the total Chern class of F and h ; i is the intersection form. This form is related to...

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 th...

We present a fast algorithm for computing the global crystal basis of the basic\(U_q (\widehat{\mathfrak{s}\mathfrak{l}}_n )\)-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 typeA at roots...

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 representation of the quantum affine algebra of type $A_{n-1}^{(1)}$ and are related to Hall-Littlewood functions via the geometry of flag varieti...

We give an expression of the $q$-analogues of the multiplicities of weights in irreducible $\sl_{n+1}$-modules in terms of the geometry of the crystal graph attached to the corresponding $U_q(\sl_{n+1})$-modules. As an application, we describe multivariate polynomial analogues of the multiplicities of the zero weight, refining Kostant's generalized...

We introduce a new family of symmetric functions, which are defined in terms of ribbon tableaux and generalize Hall-Littlewood functions. We present a series of conjectures, and prove them in two special cases. 1 Introduction Hall-Littlewood functions [Li1] are known to be related to a variety of topics in representation theory, geometry and combin...

The Hecke algebras of type An 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 defined by Hirzebruch. They provide g-idempotents, togetherwith a simple way to describe the irreducible r...

This paper presents a noncommutative theory of symmetric functions, based on the notion of quasi-determinant. We begin with a formal theory, corresponding to the case of symmetric functions in an infinite number of independent variables. This allows us to endow the resulting algebra with a Hopf structure, which leads to a new method for computing i...

This note describes ideals generated by symmetric polynomials in two sets of variables A,B. These ideals generalize the ideals generated by differences of elementary symmetric functions considered by Fischer (1988) and the ideals generated by symmetric functions in the formal difference A-B described by Pragacz (1987). The main tool is divided diff...

We give some applications of our recent work [C. R. Acad. Sci., Paris, Ser. I 316, No. 1, 1-6 (1993; Zbl 0769.05095)] about Hall-Littlewood functions at roots of unity. In particular, we prove the two conjectures of N. Sultana [Hall-Littlewood functions and their applications to representation theory, Thesis, University of Wales, 1990] on specializ...

Using divided differences 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 differences on some distinguished symmetric polynomials ( e P - polynomials) and relate it to vertex operators....