1999/07/21 by I. Grojnowski, Grojnowski, I. · 3 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #math.RT
paper · pdf · doi:10.48550/arxiv.math/9907129
arxiv created 1999/07/21 · openalex publication_date 1999/07/21 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we prove theorems that describe how the representation theory of the affine Hecke algebra of type A and of related algebras such as the group algebra of the symmetric group are controlled by integrable highest weight representations of the characteristic zero affine Lie algebra sll. In particular we parameterise the representations of these algebras by the nodes of the crystal graph, and give various Hecke theoretic descriptions of the edges. As a consequence we find for each prime p a basis of the integrable representations of sll which shares many of the remarkable properties, such as positivity, of the global crystal basis/canonical basis of Lusztig and Kashiwara. This \it p-canonical basis is the usual one when p = 0, and the crystal of the p-canonical basis is always the usual one. The paper is self-contained, and our techniques are elementary (no perverse sheaves or algebraic geometry is invoked).