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International Mathematics Research Notices (2000) 2000:413-440, doi:10.1155/S1073792800000234
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Copyright © 2000 Hindawi Publishing Corporation. All rights reserved.

The Hall algebra of a cyclic quiver and canonical bases of Fock spaces

Olivier Schiffmann

We show that the Ringel-Hall algebra Formula of the cyclic quiver of type Formula is equal to the tensor product of the (negative nilpotent) quantized enveloping algebra Formula with a polynomial algebra Formula. We use this to prove the conjecture of Varagnolo and Vasserot relating the "geometric" canonical basis of the level-1 Fock space (defined via the action of the Hall algebra) with the Leclerc-Thibon basis. In particular, this completes the approach by Varagnolo and Vasserot to a q-analogue of the Lusztig character formula for simple modules over quantum Formula at roots of unity. Finally, we generalize the previous results to the higher level Fock spaces defined by Uglov.


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