9
$\begingroup$

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

Improve this question
$\endgroup$
5
  • 3
    $\begingroup$ 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 $\endgroup$
    – Sam Hopkins
    Commented Jun 25, 2018 at 13:56
  • 1
    $\begingroup$ 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 $\endgroup$
    – Sam Hopkins
    Commented Jun 25, 2018 at 21:00
  • 2
    $\begingroup$ 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. $\endgroup$
    – Gro-Tsen
    Commented Jun 26, 2018 at 8:07
  • $\begingroup$ @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). $\endgroup$
    – Sam Hopkins
    Commented Jun 26, 2018 at 13:20
  • $\begingroup$ @SamHopkins Ah yes, thank you, I often get them backwards. $\endgroup$
    – Gro-Tsen
    Commented Jun 26, 2018 at 14:29

1 Answer 1

Reset to default
4
$\begingroup$

Young diagrams for the exceptional Lie algebras are considered in the book https://press.princeton.edu/titles/8839.html (Group Theory: Birdtracks, Lie's, and Exceptional Groups, by Predrag Cvitanovic).

Improve this answer
$\endgroup$
2
  • 2
    $\begingroup$ There is also a PDF version for download at the book's website (at the "webbook" link) birdtracks.eu . $\endgroup$
    – j.c.
    Commented Jun 26, 2018 at 8:27
  • $\begingroup$ @Zurab: Thanks! Yes all I want is in pages 209 and 210. $\endgroup$
    – Nadia SUSY
    Commented Jun 26, 2018 at 11:54

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .