Irreducible representations for the $A$-series Lie algebras are labelled Young diagrams, with a basis of each given by Young tableaux. Moreover, analogues exist for the $B,C$, and $D$ series.

Does such a description exist for the exceptional Lie algebras $$ \frak{g}_2 \subseteq \frak{f_4} \subseteq\frak{e}_6 \subseteq\frak{e}_7 \subseteq\frak{e}_8? $$

anysemisimple Lie algebra are labeled by their highest weight, which can be expressed by the (nonnegative integer) coefficients of the latter on the basis of the fundamental weights. This is what Young tableaux do (the coefficients being the differences between lengths of successive lines, or something). So if you just want tolabelrepresentations, the classical highest weight theory is all you need. If you want to branch or compute tensor products, of course, you need a more sophisticated theory. $\endgroup$Young diagramswhich index irreducible representations. Young Tableaux can be used e.g. to give a basis of the corresponding irreducible representation (since the number of tableaux of a given shape is equal to the dimension of the representation). $\endgroup$