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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  in terms of ribbon tableaux.
Let C be the category of finite-dimensional representations of a quantum affine algebra U q (g) of simply-laced type. We introduce certain monoidal subcategories C ℓ (ℓ ∈ N) of C and we study their Grothendieck rings using cluster algebras.
Noncommutative analogues of classical operations on symmetric functions are investigated , and applied to the description of idempotents and nilpotents in descent algebras. Its is shown that any sequence of Lie idempotents (one in each descent algebra) gives rise to a complete set of indecomposable orthogonal idempotents of each descent algebra, and various… (More)
Let n be a maximal nilpotent subalgebra of a complex simple Lie algebra of type A, D, E. Lusztig has introduced a basis of U (n) called the semicanonical basis, whose elements can be seen as certain constructible functions on varieties of modules over a preprojective algebra of the same Dynkin type as n. We prove a formula for the product of two elements of… (More)
A criterion of irreducibility for induction products of evaluation modules of type A affine Hecke algebras is given. It is derived from multiplicative properties of the cano-nical basis of a quantum deformation of the Bernstein-Zelevinsky ring.
We study the multiplicative properties of the dual of Lusztig's semicanonical basis. The elements of this basis are naturally indexed by the irreducible components of Lusztig's nilpotent varieties, which can be interpreted as varieties of modules over preprojective algebras. We prove that the product of two dual semicanonical basis vectors ρ Z ′ and ρ Z ′′… (More)
Let Q be a finite quiver without oriented cycles, and let Λ be the associated preprojective algebra. To each terminal CQ-module M (these are certain preinjective CQ-modules), we attach a natural subcategory CM of mod(Λ). We show that CM is a Frobenius category, and that its stable category C M is a Calabi-Yau category of dimension two. Then we develop a… (More)