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Given any group $ G $ of $ n \times n $ complex matrices there is a natural $ n $ dimensional representation of $ G $ on $ V=\mathbb{C}^n $. In this case the representation $$ V \otimes V^* \cong \mathfrak{gl}(V) $$ can be naturally viewed as $ G $ acting by conjugation on the $ n^2 $ dimensional space of all $ n \times n $ complex matrices. Whenever $ G $ is compact then one the direct summands in the decomposition of $ V \otimes V^* \cong \mathfrak{gl}(V) $ will always be the complexification of the adjoint representation of $ G $.

For example, consider the natural module $ (SU_n, V=\mathbb{C}^n) $. Then $ V \otimes V^* \cong \mathfrak{gl}(V) $ decomposes as the $ n^2-1 $ dimensional adjoint irrep $ \mathfrak{sl}_n(\mathbb{C}) $ (which is the complexification of $ \mathfrak{su}_n $) together with a 1 dimensional trivial irrep (the span of the identity matrix $ \sum_{i=1}^n e_i \otimes e_i^* $).

For the natural module $ (SO_n, V=\mathbb{C}^n) $ I'm curious about the direct sum decomposition of $ V \otimes V^* \cong \mathfrak{gl}(V) $ since it is $ n^2 $ dimensional it must include a lot of other interesting stuff beyond the $ \frac{n(n-1)}{2} $ dimensional adjoint representation.

Similarly, for the natural module $ (Sp_n, V=\mathbb{C}^{2n}) $ I'm curious about the direct sum decomposition of $ V \otimes V^* \cong \mathfrak{gl}(V) $ since it is $ 4n^2 $ dimensional it must include a lot of other interesting stuff beyond the $ n(2n+1) $ dimensional adjoint representation.

I'm sure all this is well known, so I would also be happy to accept a reference.

Ian Gershon Teixeira
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1 Answers1

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A standard reference for these results is Fulton & Harris's Representation Theory, $\S\S$16–20.

In both the orthogonal and symplectic cases, dualizing using the (respectively, symmetric and asymmetric) bilinear form preserved by the group identifies $\mathfrak{gl}(V) \cong V \otimes V^*$ with $V \otimes V$, which hence decomposes as $$\mathfrak{gl}(V) \cong \operatorname{Sym}^2 V \oplus \bigwedge\!^2\, V .$$

In the orthogonal case, $n \geq 3$, $\bigwedge\!^2 \,V$ is isomorphic to the adjoint representation, $\mathfrak{so}_n$, and $\operatorname{Sym}^2 V$ decomposes as the direct sum of (1) the line $\Bbb C g$ of scalar multiples of the symmetric bilinear form $g$ preserved by $SO_n$ and (2) the hyperplane $\operatorname{Sym}^2_\circ V$ of symmetric $2$-tensors tracefree with respect to $g$, i.e., those satisfying $g_{ab} T^{ab} = 0$. In summary, $$\mathfrak{gl}_n = \mathfrak{so}_n \oplus \operatorname{Sym}^2_\circ V \oplus \Bbb C .$$

  • For $n \neq 4$ the three summands we have identified are all irreducible.

  • For $n = 4$, $g$ and the orientation defined by $SO_4$ on $V$ together define the Hodge star endomorphism, $\ast: \bigwedge\!^2 \,V \to \bigwedge\!^2 \,V$, which satisfies $\ast^2 = \operatorname{id}$, and so $\bigwedge\!^2 \,V$ decomposes further as a direct sum of the $(\pm 1)$-eigenspaces $\bigwedge\!^2_\pm \,V$ of $\ast$, both of which are irreducible and which have dimension $3$. (That the adjoint representation is not irreducible in this case is a consequence of the fact that $SO_4$ is not simple; indeed $\mathfrak{so}_4 \cong \mathfrak{so}_3 \oplus \mathfrak{so}_3$.) We have: $$\mathfrak{gl}_4 = \overbrace{\mathfrak{so}_3 \oplus \mathfrak{so}_3}^{\mathfrak{so}_4} \oplus \operatorname{Sym}^2_\circ V \oplus \Bbb C .$$

  • For $n = 3$, the three summands are again irreducible, but notice that in this case the Hodge star operation defines an isomorphism between $\mathfrak{so}_3 \cong \bigwedge^2 V$ and $V$.

In the symplectic case the reverse is true: $\operatorname{Sym}^2 V$ is isomorphic to the adjoint representation, $\mathfrak{sp}_n$, and $\bigwedge\!^2\,V$ decomposes as the direct sum of (1) the line $\Bbb C \omega$ of scalar multiples of the symplectic form $\omega$ preserved by $Sp_n$ and (2) the hyperplane $\bigwedge\!^2{\!\!}_\circ\,V$ of skew $2$-tensors tracefree with respect to $\omega$, i.e., those satisfying $\omega_{ab} T^{ab} = 0$. So, $$\mathfrak{gl}_n \cong \mathfrak{sp}_n \oplus \bigwedge\!^2{\!\!}_\circ\,V \oplus \Bbb C .$$

Notice that for $n = 1$ all skew $2$-tensors are multiples of the dual of the symplectic form, leaving $$\mathfrak{gl}_2 \cong \mathfrak{sp}_1 \oplus \Bbb C .$$ In all cases the summands are irreducible.

Incidentally, one can carry out decompositions of semisimple representations into irreducible submodules using Lie or SAGE, both free software.

Travis Willse
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  • For the special unitary group is the adjoint representation identified with the traceless part of the second alternating power (like for the special orthogonal group) or is the adjoint representation identified with the traceless part of the second symmetric power (like for the symplectic group)? And do you know if there is anything like Lie that runs in GAP? – Ian Gershon Teixeira Oct 15 '22 at 19:22
  • If I recall correctly, the adjoint representation of $SU_n$ is identified with the tracefree, anti-Hermitian part of $V \otimes \bar V$, i.e., it's a mixture of what happens in the orthogonal and symplectic cases. (This makes sense, because a Hermitian structure on a vector spaces consists of an orthogonal structure and a compatible symplectic structure.) – Travis Willse Oct 15 '22 at 22:14
  • I don't know GAP well, so I don't know whether it can solve "branching" problems like this one. – Travis Willse Oct 15 '22 at 22:15