Séminaire Lotharingien de Combinatoire, 82B.2 (2019), 12 pp.

Il-Seung Jang and Jae-Hoon Kwon

Quantum nilpotent subalgebras of classical quantum groups and affine crystals

Abstract. We give a new interpretation of RSK correspondence of type D in terms of affine crystals. We show that the crystal of quantum nilpotent subalgebra of U_q(D_n) associated to a maximal Levi subalgebra of type A_n-1 has an affine D_n⁽¹⁾-crystal structure, and it is isomorphic to a direct limit of perfect Kirillov-Reshetikhin crystal B^n,s for s >= 1. An analogue of RSK correspondence for type D due to Burge is naturally defined on this crystal and shown to be an isomorphism of affine crystals. We further obtain a generalization of Greene's formula for type D and as a byproduct a new polytope realization of B^n,s.

Received: November 15, 2018. Accepted: February 17, 2019. Final version: April 1, 2019.

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