A Littlewood-Richardson rule for evaluation representations of quantum affine sl(n)

We give a combinatorial description of the composition factors of the induction product of two evaluation modules of the affine Iwahori-Hecke algebra of type GL(m). Using quantum affine Schur-Weyl duality, this yields a combinatorial description of the composition factors of the tensor product of two evaluation modules of the quantum affine algebra of type sl(n)

Cluster algebras and representation theory

We apply the new theory of cluster algebras of Fomin and Zelevinsky to study some combinatorial problems arising in Lie theory. This is joint work with Geiss and Schr\"oer (3, 4, 5, 6), and with Hernandez (8, 9)

Constructible characters and canonical bases

We give closed formulas for all vectors of the canonical basis of a level 2 irreducible integrable representation of $U_v(sl_\infty)$. These formulas coincide at v=1 with Lusztig's formulas for the constructible characters of the Iwahori-Hecke algebras of type B and D.Comment: 16 page

Monoidal categorifications of cluster algebras of type A and D

In this note, we introduce monoidal subcategories of the tensor category of finite-dimensional representations of a simply-laced quantum affine algebra, parametrized by arbitrary Dynkin quivers. For linearly oriented quivers of types A and D, we show that these categories provide monoidal categorifications of cluster algebras of the same type. The proof is purely representation-theoretical, in the spirit of [arXiv:0903.1452].Comment: 15 pages ; to appear in the proceedings of the Conference Symmetries, Integrable systems and Representation

Nakajima varieties and repetitive algebras

We realize certain graded Nakajima varieties of finite Dynkin type as orbit closures of repetitive algebras of Dynkin quivers. As an application, we obtain that the perverse sheaves introduced by Nakajima for describing irreducible characters of quantum loop algebras are isomorphic to the intersection cohomology sheaves of these orbit closures.Comment: 26 pages, v2 : Minor corrections. Final version to appear in PRIMS (Publications of the Research Institute for Mathematical Sciences, Kyoto

Aggregation and residuation

In this paper, we give a characterization of meet-projections in simple atomistic lattices that generalizes results on the aggregation of partitions in cluster analysis.Aggregation theory ; dependence relation ; meet projection ; partition ; residual map ; simple lattice