Crystal of affine type \(\widehat{A}_{\ell-1}\) and Hecke algebras at a primitive \(2\ell\) th root of unity. (English) Zbl 1497.20008
Let \(\ell\ge 2\) be an integer. In this paper, the authors provide a realization of the crystal of affine type \(\widehat{A}_{\ell-1}\) using the modular representation theory of the affine Hecke algebras \(\mathcal{H}_n\) of type \(A\) and their level two cyclotomic quotients whose Hecke parameter is a primitive 2\(\ell\)th root of unity. The main results of this paper generalize earlier work of I. Grojnowski and M. Vazirani on the relations between the crystal of \(\widehat{\mathfrak{sl}}_\ell\) and the affine Hecke algebras of type \(A\) at a primitive \(\ell\)th root of unity [Transform. Groups 6, No. 2, 143–155 (2001; Zbl 1056.20002)].
Reviewer: Frauke Bleher (Iowa City)
MSC:
| 20C08 | Hecke algebras and their representations |
| 17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |
Citations:
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