2001/07/06 by Monica Vazirani, M. Vazirani, Vazirani, M.
Mathematics · #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT) #math.CO #math.QA #math.RT
paper · pdf · doi:10.48550/arxiv.math/0107052
33 pages
arxiv created 2001/07/06 · openalex publication_date 2001/07/06 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \it 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.