Lusztig data of Kashiwara–Nakashima tableaux in types B and C

Abstract

We provide an explicit combinatorial description of the embedding of the crystal of Kashiwara-Nakashima tableaux in types B and C into that of i-Lusztig data associated to a family of reduced expressions i of the longest element w0. The description of the embedding is simple and elementary using only the Schützenberger's jeu de taquin and RSK algorithm. A spinor model for classical crystals plays an important role as an intermediate object connecting Kashiwara-Nakashima tableaux and Lusztig data.2010 Mathematics Subject Classification. 17B37, 22E46, 05E10.(1.2)where r " xω λ , h n y, T λ is another tableau model of Bpω λ q introduced in [19,20], and V λ is the crystal of a parabolic Verma module with respect to a maximal Levi subalgebra l of type A n´1 in g [16,17].

Quantum nilpotent subalgebras of classical quantum groups and affine crystals

“…Let u ´be the negative nilradical of the parabolic subalgebra p " l `b, where b is a Borel subalgebra of g. The enveloping algebra U pu ´q is an integrable l-module, which has a multiplicity-free decomposition [9], and the expansion of its character (1.1) ch U pu ´q " ź D p2q n`1 and C p1q n when g is of type B n and C n , respectively. It is done by regarding the set of biwords (or the set of matrices with non-negative integral entries) as the crystal BpU q pu ´qq of the quantum nilpotent subalgebra U q pu ´q (see also [20,21]). This approach enables us to define naturally the Kashiwara operators on both sides of the correspondence κ for the simple roots other than the ones in l. Moreover it is proved that BpU q pu ´qq is isomorphic to a limit of perfect Kirillov-Reshetikhin crystals B r,s , which are classically irreducible (cf.…”