Consider an irreducible representation $R$ of a simple compact Lie group $G$, such that $R$ is isomorphic to its dual. This isomorphism (unique up to scaling by Schur's lemma) defines a trivial one-dimensional subrepresentation of $R\otimes R=S^2R\oplus\Lambda^2R$ and the representation is called real/quaternionic if $S^2R$ (resp $\Lambda^2R$) contains this one-dimensional representation of $G$.

I am interested in the multiplicities $p_S$ and $p_\Lambda$ with which the adjoint representation of $G$ appears in the decompositions of $S^2R$ and $\Lambda^2R$ into irreducible representations. Experimentally, I observed that

• $p_\Lambda \geq p_S$ for a real representation,

• $p_S > p_\Lambda$ for a quaternionic representation.

Is it true? Here are some useless facts I have derived so far.

• For a real representation, $p_\Lambda\geq 1$. For a quaternionic representation, $p_S\geq 1$. (This comes from looking at the action of $\text{Lie}(G)$ on $R$ as an element of $\text{Hom}(\text{adj}_G\otimes R,R) \simeq \text{Hom}(\text{adj}_G, R\otimes R)$ and tracking symmetries.)

• $p=p_\Lambda+p_S$ is equal to the number of simple roots $\alpha$ of $G$ such that $\langle\alpha,\mu\rangle>0$, where $\mu$ is the highest weight of $R$. In other words it is equal to the rank of $G$ minus the number of boundaries of the Weyl chamber that $\mu$ sits on. Any counterexample must have $p\geq 2$, which makes it a bit hard to scan for counterexamples.

• The question can be reformulated in terms of the Frobenius-Schur indicator and a variant involving the character $\chi_{\text{adj}}(g)$, namely I claim that $\int_{g\in G}\chi(g^2)d\mu$ and $\int_{g\in G}\chi_{\text{adj}}(g) \chi(g^2)d\mu$ have opposite sign, where $\mu$ is the Haar measure.

This came up in the course of writing an article on supersymmetric gauge theories. I can weaken our conclusions slightly to avoid needing this result, but I would prefer to find a proof or counterexample.

EDIT (November 1, 2022): Over the weekend I think I found a technique to determine the exact multiplicities, according to how conjugation acts on the fundamental weights. While I haven't done the precise counting yet, the inequalities I was suggesting here are indeed correct. I plan to write it up with a colleague in the next couple of weeks and post an answer to this question here as soon as I can manage.

Consider an irreducible representation $R$ of a simple compact Lie group $G$, such that $R$ is isomorphic to its dual. This isomorphism (unique up to scaling by Schur's lemma) defines a trivial one-dimensional subrepresentation of $R\otimes R=S^2R\oplus\Lambda^2R$ and the representation is called real/quaternionic if $S^2R$ (resp $\Lambda^2R$) contains this one-dimensional representation of $G$.

I am interested in the multiplicities $p_S$ and $p_\Lambda$ with which the adjoint representation of $G$ appears in the decompositions of $S^2R$ and $\Lambda^2R$ into irreducible representations. Experimentally, I observed that

• $p_\Lambda \geq p_S$ for a real representation,

• $p_S > p_\Lambda$ for a quaternionic representation.

Is it true? Here are some useless facts I have derived so far.

• For a real representation, $p_\Lambda\geq 1$. For a quaternionic representation, $p_S\geq 1$. (This comes from looking at the action of $\text{Lie}(G)$ on $R$ as an element of $\text{Hom}(\text{adj}_G\otimes R,R) \simeq \text{Hom}(\text{adj}_G, R\otimes R)$ and tracking symmetries.)

• $p=p_\Lambda+p_S$ is equal to the number of simple roots $\alpha$ of $G$ such that $\langle\alpha,\mu\rangle>0$, where $\mu$ is the highest weight of $R$. In other words it is equal to the rank of $G$ minus the number of boundaries of the Weyl chamber that $\mu$ sits on. Any counterexample must have $p\geq 2$, which makes it a bit hard to scan for counterexamples.

• The question can be reformulated in terms of the Frobenius-Schur indicator and a variant involving the character $\chi_{\text{adj}}(g)$, namely I claim that $\int_{g\in G}\chi(g^2)d\mu$ and $\int_{g\in G}\chi_{\text{adj}}(g) \chi(g^2)d\mu$ have opposite sign, where $\mu$ is the Haar measure.

This came up in the course of writing an article on supersymmetric gauge theories. I can weaken our conclusions slightly to avoid needing this result, but I would prefer to find a proof or counterexample.