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

We construct every finite-dimensional irreducible representation of the simple Lie algebra of type \({\textsf {E}}_{7}\) whose highest weight is a nonnegative integer multiple of the dominant minuscule weight associated with the type \({\textsf {E}}_{7}\) root system. As a consequence, we obtain constructions of each finite-dimensional irreducible representation of the simple Lie algebra of type \({\textsf {E}}_{6}\) whose highest weight is a nonnegative integer linear combination of the two dominant minuscule \({\textsf {E}}_{6}\)-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 \({\textsf {E}}_{6}\)- and \({\textsf {E}}_{7}\)-polyminuscule lattices that analogize certain lattices associated with the famous special linear Lie algebra representation constructions obtained by Gelfand and Tsetlin.