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

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