Alain Lascoux

Hecke algebras at roots of unity and crystal bases of quantum affine algebras

AbstractWe present a fast algorithm for computing the global crystal basis of the basic

Tableau atoms and a new Macdonald positivity conjecture

U_q (\widehat{\mathfrak{s}\mathfrak{l}}_n )

Crystal graphs andq-analogues of weight multiplicities for the root systemA 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 of unity. We conjecture that, upon specializationq→1, our algorithm computes the decomposition matrices of all Hecke algebras at anth root of 1.

The singular locus of a Schubert variety

We introduce a new family of symmetric functions, which are q analogs of products of Schur functions, defined in terms of ribbon tableaux. These functions can be interpreted in terms of the Fock space representation Fq of Uq(sln), and are related to Hall–Littlewood functions via the geometry 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 Fq using ribbon tableaux.

Anneau de Grothendieck de la variété de drapeaux

A snap action fluid control valve, the operation of which is controlled by a relatively slow acting thermally responsive actuator member. The valve of this invention is particularly adapted for use in controlling flow of fluid to a fluid operable clutch or the like for operation thereof. The thermally responsive actuator portion of the valve senses temperature of a fluid, the temperature of which is responsive to operation of a fan which is operated through the clutch.

Ribbon Schur functions

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).

SYMMETRICA, an object oriented computer-algebra system for the symmetric group

AbstractWe give an expression of theq-analogues of the multiplicities of weights in irreducible