Parameterizing Hecke algebra modules: Bernstein-Zelevinsky multisegments, Kleshchev multipartitions, and crystal graphs

Original · EN

This paper provides a combinatorial dictionary between three sets of objects: Bernstein-Zelevinsky multisegments, Kleshchev multipartitions, and the irreducible modules of the affine Hecke algebra Hₙ (for generic q). In particular, we compute the action of the crystal operator eᵢ (a refinement of socle of Restriction) on an irreducible module both in terms of its parameterization by multisegments and by multipartitions. In other words, we give explicit crystal graph isomorphisms. A byproduct is the determination of which multisegments parameterize modules of the cyclotomic Hecke algebra Hₙλ. The theorems also explain why the rule for computing eᵢ mirrors the rule we know for that on a tensor product of crystal graphs. We also give a construction of the irreducible module parameterized by a multipartition without relying on a choice of path in the crystal graph. The proofs given here are elementary and do not rely on any geometry.