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

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 $\mathsf{B}(\infty)$.