The cycle enumerator of unimodal permutations

We give a generating function for the number of unimodal permutations with a given cycle structure.Comment: 7 pages, LaTe

Canonical bases of q-deformed Fock spaces

We define a canonical basis of the $q$-deformed Fock space representation of the affine Lie algebra \glchap_n. We conjecture that the entries of the transition matrix between this basis and the natural basis of the Fock space are $q$-analogues of decomposition numbers of the $v$-Schur algebras for $v$ specialized to a $n$th root of unity.Comment: 7 pages, LaTe

Littlewood-Richardson coefficients and Kazhdan-Lusztig polynomials

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 multiplicities coincide with those previously introduced in terms of ribbon tableaux.Comment: 51 pages, 10 Figures, Minor corrections and typos, and some new comments concerning Soergel's character formula for tilting module

Higher order peak algebras

Using the theory of noncommutative symmetric functions, we introduce the higher order peak algebras, a sequence of graded Hopf algebras which contain the descent algebra and the usual peak algebra as initial cases (N = 1 and N = 2). We compute their Hilbert series, introduce and study several combinatorial bases, and establish various algebraic identities related to the multisection of formal power series with noncommutative coefficients.Comment: 20 pages, AMS LaTe