# First Explicit Irreducible Representations

Although the classification of simple Lie Algebras and their representations is fully understood, I wonder whether there is some book with exhaustive tables describing explicit irreducible representations in low dimensions. For instance, it would be very helpful for me to know which are the first irreducible representations of $\mathfrak{so}(7)$ and $\mathfrak{so}(8)$ (up to dimension 30, say).

• What sort of description are you looking for? – Tobias Kildetoft Mar 11 '15 at 11:09
• @TobiasKildetoft Something like '$\mathfrak{so}(7)$: In dim. 1, just the trivial representation. In dim. 7, we have 3 (say) inequivalent representations induced by...'. – Jjm Mar 11 '15 at 11:16
• How would the "induced by" be continued? – Tobias Kildetoft Mar 11 '15 at 11:18
• @TobiasKildetoft Well, only if it may be easily described. Vincent's proposal is quite the approach i was looking for. – Jjm Mar 11 '15 at 11:23

I think "Group Theory for Unified Model Building" by R. Slansky qualifies. As the title suggests it is written with an application in physics (beyond my understanding) in mind, but the tables are very useful for purely mathematical purposes as well. (Disclaimer: it is quite some years ago that I last read it.)

If all you really want to know are the irreducible representations of ${\frak{so}}(7)$ and ${\frak{so}}(8)$ up to dimension $30$, I can save you the trouble of looking up these tables:

For ${\frak{so}}(7)$, there are the following irreducible representations with dimensions below $30$:

1. $\mathbb{R}^1$ (the trivial representation)
2. $\mathbb{R}^7$ (the 'vector' or 'standard' representation)
3. $\mathbb{R}^8$ (the 'spinor' representation)
4. $\mathbb{R}^{21}$ (the 'adjoint' representation, i.e., 7-by-7 skew-symmetric matrices)
5. $\mathbb{R}^{27}$ (the traceless symmetric 7-by-7 matrices, i.e., $S^2_0(\mathbb{R}^7)$)

Of course, you probably really also want a couple more, such as

1. $\mathbb{R}^{35}$ (traceless symmetric 8-by-8 matrices, i.e., $S^2_0(\mathbb{R}^8)$; also equals $\Lambda^3(\mathbb{R}^7)$)
2. $\mathbb{R}^{48}$ (the 'other' irreducible component in $\mathbb{R}^7\otimes \mathbb{R}^8 \simeq \mathbb{R}^8 \oplus \mathbb{R}^{48}$)

For ${\frak{so}}(8)$, there are the following irreducible representations with dimensions below $30$:

1. $\mathbb{R}^1$ (the trivial representation)
2. $V = \mathbb{R}^8$ (the 'vector' or 'standard' representation)
3. $S_+ = \mathbb{R}^8$ (the 'plus spinor' representation)
4. $S_- = \mathbb{R}^8$ (the 'minus spinor' representation))
5. $\Lambda^2(V) = \Lambda^2(S_+) = \Lambda^2(S_-) = \mathbb{R}^{28}$ (the skew-symmetric 8-by-8 matrices)

But, you might want a few more, such as

1. $S^2_0(V) = \mathbb{R}^{35}$ (traceless symmetric 8-by-8 matrices)
2. $S^2_0(S_+) = \mathbb{R}^{35}$ (traceless symmetric 'plus spinor' squares)
3. $S^2_0(S_-) = \mathbb{R}^{35}$ (traceless symmetric 'minus spinor' squares)

(Note that $\Lambda^4(V) = \Lambda^4_+(V)\oplus \Lambda^4_-(V) = S^2_0(S_+)\oplus S^2_0(S_-)$, while $\Lambda^4(S_+) = S^2_0(S_-)\oplus S^2_0(V)$, etc.)

1. $\Lambda^3(V) = \mathbb{R}^{56}$ (vector $3$-forms = 'other' component in $S_+\otimes S_- = V \oplus \Lambda^3(V)$)
2. $\Lambda^3(S_+) = \mathbb{R}^{56}$ (plus spinor $3$-forms = 'other' component in $S_-\otimes V = S_+ \oplus \Lambda^3(S_+)$)
3. $\Lambda^3(S_-) = \mathbb{R}^{56}$ (minus spinor $3$-forms = 'other' component in $S_+\otimes V = S_- \oplus \Lambda^3(S_-)$)

That should be enough to get you started, including knowing the irreducible decompositions of the exterior powers of the three $8$-dimensional representations.

The next smallest irreducible has dimension 112.

• Thank you very much, really. There is a big effort in your answer, and for sure it will be most helpful for me. Thanks!! – Jjm Mar 14 '15 at 14:26
• @Robert Bryant Was this from the top of your head? If not where can one find this kind of data. I know where to find tables which give dimensions and weights but this kind of data (i.e. using as much linear algebraic operations $\bigwedge^n, S^n$ decomposition of these etc...\$ is much more helpful from a theoretical perspective. The only way I had success with aquiring such descriptions was by computing by myself small examples. – Saal Hardali Jun 18 '17 at 11:01
• @SaalHardali: Yes, that was from memory. From time to time, I've had to use details of the representation theory of these two algebras in my work in differential geometry, and the above are some of the facts about them that I learned and have found useful again and again. – Robert Bryant Jun 23 '17 at 14:13
• it should be straight forward to automate finding such expressions : GAP for example can construct all these representations explicitly and can give decompositions of symmetric and exterior products.. – unknown Jun 23 '17 at 17:41

Besides Slansky's paper (1981) there are also