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The Hall algebra of a cyclic quiver and canonical bases of the Fock space

Olivier Schiffmann
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Abstract

We prove that the Hall algebra U^-_n of the cyclic quiver of type A^(1)_{n-1} decomposes as a direct product of the quantum negative nilpotent subalgebra U_q^-(\hat{sl}_{n)) and C[q,q^{-1},z_1,z_2...]. We use this to prove a conjecture of Varagnolo and Vasserot in math/9803023 relating the canonical basis of U^-_n and the canonical basis of the level 1 Fock space representation of U_q(\hat{sl}_n) introduced by Leclerc and Thibon. This yields a proof of the positivity conjecture of Lascoux, Leclerc and Thibon, and a q-analogue of the Lusztig character formula for simple modules of the quantum group U_q(sl_n) at a root of unity.

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Olivier Schiffmann. 1999-09-08. The Hall algebra of a cyclic quiver and canonical bases of the Fock space. https://arxiv.org/abs/math/9906069

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