Explicit constructions of some infinite families of finite-dimensional irreducible representations of the type $\mathsf{E}_{6}$ and $\mathsf{E}_{7}$ simple Lie algebras

Abstract

We construct every finite-dimensional irreducible representation of the simple Lie algebra of type $\mathsf{E}_{7}$ whose highest weight is a nonnegative integer multiple of the dominant minuscule weight associated with the type $\mathsf{E}_{7}$ root system. As a consequence, we obtain constructions of each finite-dimensional irreducible representation of the simple Lie algebra of type $\mathsf{E}_{6}$ whose highest weight is a nonnegative integer linear combination of the two dominant minuscule $\mathsf{E}$-weights. Our constructions are explicit in the sense that, if the representing space is $d$-dimensional, then a weight basis is provided such that all entries of the $d \times d$ representing matrices of the Chevalley generators are obtained via explicit, non-recursive formulas. To effect this work, we introduce what we call $\mathsf{E}_{6}$- and $\mathsf{E}_{7}$-polyminuscule lattices that analogize certain lattices associated with the famous special linear Lie algebra representation constructions obtained by Gelfand and Tsetlin.