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AbstractWe 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… (More)

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… (More)

We present representation theoretical interpretations ofquasi-symmetric functions and noncommutative symmetric functions in terms ofquantum linear groups and Hecke algebras at q = 0. We obtain inthis… (More)

We show that the Littlewood-Richardson coefficients are values at 1 of certain parabolic Kazhdan-Lusztig polynomials for affine symmetric groups. These q-analogues of Littlewood-Richardson… (More)

We introduce a graded Hopf algebra based on the set of parking functions (hence of dimension (n + 1) in degree n). This algebra can be embedded into a noncommutative polynomial algebra in infinitely… (More)

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… (More)

Using the theory of noncommutative symmetric functions, we introduce the higher order peak algebras (Sym(N))N≥1, a sequence of graded Hopf algebras which contain the descent algebra and the usual… (More)

We use the Hopf algebra structure of the algebra of symmetric functions to study the Adams operators of the complex representation rings of symmetric groups, and we give new proofs of all of… (More)

A special family of partitions occurs in two apparently unrelated contexts: the evaluation of one-dimensional configuration sums of certain RSOS models, and the modular representation theory of… (More)