Affine sl_p controls the representation theory of the symmetric group and related Hecke algebras

arXivmath/9907129MaRDI QIDQ6501414

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Abstract: 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 hat{sl}_l. 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 hat{sl}_l 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).

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