Descriptions of crystal B(λ) for E6 and E7 types via tableaux and Kashiwara embedding

Abstract

Let Uq(g) be the quantized universal enveloping algebra for a Lie algebra g, and let Vq(λ) be the irreducible highest weight module for Uq(g). The crystal base B(λ) is a colored directed graph that captures the structure of Vq(λ) and the action of Uq(g) on Vq(λ) in a rudimentary manner. Likewise, the crystal base B(∞) holds the bare structure of the negative part Uq−(g). In this work, we describe realizations of the crystal B(λ) via two separate approaches for the cases when the base Lie algebra g is of E6 and E7 types. Our first description relies on the fact that B(λ) appears as a connected component within the much larger crystal B(∞)⊗{rλ}, where {rλ} is a certain single-element crystal. Choosing to represent elements of B(∞) with marginally large tableaux, we identify those elements belonging to the mentioned connected component. Our second description of B(λ) is a translation of the first description into one involving the Kashiwara embedding, which is an embedding of B(∞) into a tensor product of a series of much simpler crystals.