Bernard Leclerc

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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 Λ be a preprojective algebra of simply laced Dynkin type ∆. We study maximal rigid Λ-modules, their endomorphism algebras and a mutation operation on these modules. This leads to a representation-theoretic construction of the cluster algebra structure on the ring C[N ] of polynomial functions on a maximal unipotent subgroup N of a complex Lie group of(More)
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 ρZ′ and ρZ′′ is(More)
We give a combinatorial algorithm for computing Zelevinsky’s involution of the set of isomorphism classes of irreducible representations of the affine Hecke algebra Ĥm(t) when t is a primitive nth root of 1. We show that the same map can also be interpreted in terms of aperiodic nilpotent orbits of Z/nZ-graded vector spaces.
Let n be the maximal nilpotent subalgebra of a simple complex Lie algebra g. We introduce the notion of imaginary vector in the dual canonical basis of Uq(n), and we give examples of such vectors for types An(n > 5), Bn(n > 3), Cn(n > 3), Dn(n > 4), and all exceptional types. This disproves a conjecture of Berenstein and Zelevinsky about q-commuting(More)
We give an expression of the q-analogues of the multiplicities of weights in irreducible sl n+1-modules in terms of the geometry of the crystal graph attached to the corresponding Uq(sl n+1)-modules. As an application, we describe multivariate polynomial analogues of the multiplicities of the zero weight, refining Kostant’s generalized exponents.