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

TL;DR

This paper provides explicit constructions of certain finite-dimensional irreducible representations of the simple Lie algebras of types E6 and E7, using novel lattice structures and explicit matrix formulas.

Contribution

It introduces E6- and E7-polyminuscule lattices and explicit matrix formulas for representations, extending Gelfand-Tsetlin type constructions to these exceptional Lie algebras.

Findings

Explicit matrix formulas for E6 and E7 representations

Introduction of E6- and E7-polyminuscule lattices

Construction of all representations with specific highest weights

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.