First Explicit Irreducible Representations

Question

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).

Answer 1

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.)

Answer 2

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

6. $\mathbb{R}^{35}$ (traceless symmetric 8-by-8 matrices, i.e., $S^2_0(\mathbb{R}^8)$; also equals $\Lambda^3(\mathbb{R}^7)$)
7. $\mathbb{R}^{48}$ (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

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

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

Answer 3

Besides Slansky's paper (1981) there are also

• Tits, Jacques Tabellen zu den einfachen Lie Gruppen und ihren Darstellungen (1967);

• McKay, W. G.; Patera, J. Tables of dimensions, indices, and branching rules for representations of simple Lie algebras (1981);

• McKay, W. G.; Patera, J.; Rand, D. W. Tables of representations of simple Lie algebras. Vol. I. Exceptional simple Lie algebras (1990).