Young wall realizations of level 1 irreducible highest weight and Fock space crystals of quantum affine algebras in type E

Abstract

We construct Young wall models for the crystal bases of level 1 irreducible highest weight representations and Fock space representations of quantum affine algebras in types $E_6^{\left(1\right)}$, $E_7^{\left(1\right)}$ and $E_8^{\left(1\right)}$. In each case, Young walls consist of coloured blocks stacked inside the relevant Young wall pattern which satisfy a certain combinatorial condition. Moreover the crystal structure is described entirely in terms of adding and removing blocks.