Return to Revisions

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)

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

6. $\mathbb{R}^{35}$ (traceless symmetric 8-by-8 matrices)
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.