# Young tableaux for exceptional Lie algebras

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 $$\frak{g}_2 \subseteq \frak{f_4} \subseteq\frak{e}_6 \subseteq\frak{e}_7 \subseteq\frak{e}_8?$$

• I am not an expert but I think the "Littelmann path model" is something sort of like what you are asking about: en.wikipedia.org/wiki/Littelmann_path_model – Sam Hopkins Jun 25 '18 at 13:56
• These slides seem like a nice introduction to the Littelmann path model theory, and explain the connection with tableaux as well: people.bath.ac.uk/lpah20/GeomSemNP.pdf – Sam Hopkins Jun 25 '18 at 21:00
• Maybe it's worth pointing out the "well known": irreducible representations of any semisimple 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 to label representations, 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. – Gro-Tsen Jun 26 '18 at 8:07
• @Gro-Tsen: it's Young diagrams which 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). – Sam Hopkins Jun 26 '18 at 13:20
• @SamHopkins Ah yes, thank you, I often get them backwards. – Gro-Tsen Jun 26 '18 at 14:29