Young Wall Realization of Crystal Graphs for U q (C n (1) )

Abstract

Abstract. We give a realization of crystal graphs for basic representations of the quantum affine algebra Uq(C (1) n ) using combinatorics of Young walls. The notion of splitting blocks plays a crucial role in the construction of crystal graphs.

Fock space representations and crystal bases for Cn(1)

“…Various Fock space representations and crystal bases have been constructed in [10] for other affine Kac-Moody Lie algebras, and some of these results have been used in [11] to describe Fock spaces and crystal graphs of the Lie algebra of type A (2) 2n in terms of coloured Young diagrams. In [4,8] and [7], the crystal graphs for some representations (those with dominant integral highest weights of level 1) of the quantized universal enveloping algebras of the affine Lie algebras of types A (2) 2n−1 , n 3, D (1) n , n 4, A (2) 2n , n 2, D (2) n+1 , n 2, and B (1) n , n 3, were described using new combinatorial objects called Young walls, which are built out of cubes and "half-cubes". Using Young walls to obtain the results for the algebra of type C (1) n proved to be more difficult.…”

“…In [4,8] and [7], the crystal graphs for some representations (those with dominant integral highest weights of level 1) of the quantized universal enveloping algebras of the affine Lie algebras of types A (2) 2n−1 , n 3, D (1) n , n 4, A (2) 2n , n 2, D (2) n+1 , n 2, and B (1) n , n 3, were described using new combinatorial objects called Young walls, which are built out of cubes and "half-cubes". Using Young walls to obtain the results for the algebra of type C (1) n proved to be more difficult. In [5] and [6], the authors obtain the results for g of type C (1) 2 and in [1] they obtain a description of the crystal graphs for C (1) n using Young walls.…”

Crystals of type $D_n^{(1)}$ and Young walls

“…Readers familiar with previous Young wall theory [4,10] may recall that the splitting of a block was defined to be the breaking off of the covering block in the C (1) n , A (2) 2n , and D (2) n+1 cases and the breaking off of the supporting block in the B (1) n and A (2) 2n−1 cases. Likewise, it is possible to develop the rest of the theory of Young walls for type D (1) n after defining the splitting of a block to be just one of the two possible choices.…”

“…We first introduce the notions of splitting of blocks and slices, and give a new realization of level-l perfect crystals as the equivalence classes of slices. The two notions presented in this work are refined versions of those appearing in previous works [4] and [10]. We then proceed to define the notion of Young walls, proper Young walls, reduced proper Young walls, and ground-state walls.…”

“…The C (1) n case, which was missing from these works, was more difficult to deal with, because the level-1 perfect crystal for this type is intrinsically of level-2. The difficulty was resolved in [4] by introducing the notions of slices and splitting of blocks. It was also in this work that a realization of perfect crystals, obtained in the process of Young wall realization, was explicitly stated as a separate result.…”