Parameterizing Hecke Algebra Modules: Bernstein-Zelevinsky Multisegments, Kleshchev Multipartitions, and Crystal Graphs
This paper provides a combinatorial and representation-theoretic dictionary between three sets of objects: multisegments, multipartitions, and the irreducible modules of the affine Hecke algebra H n of type A (for generic q). The dictionary is dictated by Grojnowski's theorem [G] that gives a crystal structure on the irreducible modules of H n ,n ≥ 0. (See also [A2], [AM].) Given a Kleshchev multipartition, we construct an irreducible module as the unique simple quotient of a module induced up from smaller Hecke algebras. We compare that to Bernstein-Zelevinsky's [BZ, Z] construction of an irreducible module given a multisegment. We describe the corresponding crystal structure on multisegments and multipartitions. However, the new result here is not the explicit description of the crystal graph, but the identification of three different constructions of the irreducible modules, all of which give you the same graph when the edges are given by taking simple submodules in a restriction.