%0 Generic %1 donnelly2021explicit %A Donnelly, Robert G. %A Dunkum, Molly W. %A White, Austin %D 2021 %K Represenations algebras lie %T Explicit constructions of some infinite families of finite-dimensional irreducible representations of the type $E_6$ and $E_7$ simple Lie algebras %U http://arxiv.org/abs/2109.02835 %X We construct every finite-dimensional irreducible representation of the simple Lie algebra of type $E_7$ whose highest weight is a nonnegative integer multiple of the dominant minuscule weight associated with the type $E_7$ root system. As a consequence, we obtain constructions of each finite-dimensional irreducible representation of the simple Lie algebra of type $E_6$ whose highest weight is a nonnegative integer linear combination of the two dominant minuscule $E_6$-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 $d d$ representing matrices of the Chevalley generators are obtained via explicit, non-recursive formulas. To effect this work, we introduce what we call $E_6$- and $E_7$-polyminuscule lattices that analogize certain lattices associated with the famous special linear Lie algebra representation constructions obtained by Gelfand and Tsetlin.

@misc{donnelly2021explicit, abstract = {We construct every finite-dimensional irreducible representation of the simple Lie algebra of type $\mbox{E}_{7}$ whose highest weight is a nonnegative integer multiple of the dominant minuscule weight associated with the type $\mbox{E}_{7}$ root system. As a consequence, we obtain constructions of each finite-dimensional irreducible representation of the simple Lie algebra of type $\mbox{E}_{6}$ whose highest weight is a nonnegative integer linear combination of the two dominant minuscule $\mbox{E}_{6}$-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 $\mbox{E}_{6}$- and $\mbox{E}_{7}$-polyminuscule lattices that analogize certain lattices associated with the famous special linear Lie algebra representation constructions obtained by Gelfand and Tsetlin.}, added-at = {2022-05-11T10:35:40.000+0200}, author = {Donnelly, Robert G. and Dunkum, Molly W. and White, Austin}, biburl = {https://www.bibsonomy.org/bibtex/29adaef3665cc6d9fe54dd8eb5863de50/dragosf}, description = {Explicit constructions of some infinite families of finite-dimensional irreducible representations of the type $\mbox{E}_{6}$ and $\mbox{E}_{7}$ simple Lie algebras}, interhash = {d32f27246994392ce073ce0af129177c}, intrahash = {9adaef3665cc6d9fe54dd8eb5863de50}, keywords = {Represenations algebras lie}, note = {cite arxiv:2109.02835Comment: v2: includes minor edits and updates of some references; 28 pages including 11 pages of figures}, timestamp = {2022-05-11T10:35:40.000+0200}, title = {Explicit constructions of some infinite families of finite-dimensional irreducible representations of the type $\mbox{E}_{6}$ and $\mbox{E}_{7}$ simple Lie algebras}, url = {http://arxiv.org/abs/2109.02835}, year = 2021 }