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 $E_6^{(1)}$, $E_7^{(1)}$ and $E_8^{(1)}$. 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.

51 pages, comments very welcome; v2: minor edits; v3: minor correction to §3.2.2 + extended Prop. 3.4 to all affine types