Crystal ℬ ( λ ) $\mathcal {B}(\lambda )$ as a Subset of the Tableau Description of ℬ ( ∞ ) $\mathcal {B}(\infty )$ for the Classical Lie Algebra Types

Abstract

Abstract. We study the crystal base of the negative part of a quantum group. An explicit realization of the crystal is given in terms of Young tableaux for types An, Bn, Cn, Dn, and G 2 . Connection between our realization and a previous realization of Cliff is also given.

Description of B ( ∞ ) $\mathsf {B}(\infty )$ through Kashiwara Embedding for Lie Algebra Types E 6 and E 7

“…Recall that the circled elements in the first diagram of Appendix A.1 are the nodes ofC 5 . For each group of arrows, there is precisely one element ofC 5 that is positioned at the head of one of the arrows.…”

“…This definition appeared previously in [5] for the classical Lie algebra types, with the small extension that set t r,c to infinity for the source node c of C r . If c ∈ C r appears in one of the boxes on the r-th row of T , then t r,c is the number of all boxes containing c and all the boxes appearing to their right.…”