Description of $\mathsf{B}(\infty)$ through Kashiwara Embedding for $E_6$ and $E_7$ Lie Algebra Types

TL;DR

This paper provides an explicit description of the crystal base () for the quantum groups of types E6 and E7 using Kashiwara embedding, linking it to tableau representations.

Contribution

It introduces a detailed Kashiwara embedding for () in E6 and E7, connecting it with tableau descriptions for the first time.

Findings

Explicit Kashiwara embedding for E6 and E7 types.

Connection established between embedding and tableau description.

Enhanced understanding of crystal bases for complex Lie types.

Abstract

We study the crystal base $\mathsf{B}(\infty)$ associated with the negative part of the quantum group for finite simple Lie algebras of types $E_6$ and $E_7$. We present an explicit description of $\mathsf{B}(\infty)$ as the image of a Kashiwara embedding that is in natural correspondence with the tableau description of $\mathsf{B}(\infty)$.