Adjoints in symmetric squares of Lie algebra representations

Abstract

*Caveat: we learned post-factum that most of these results are not novel. We are keeping this paper for continuity reasons.* Given finite-dimensional complex representations $V$ and $V^{\prime }$ of a simply-connected semisimple compact Lie group $G$, we determine the dimension of the $G$-invariant subspace of $\mathrm{adj}(G)\otimes V\otimes V^{\prime }$, of $\mathrm{adj}(G)\otimes S^{2}V$, and of $\mathrm{adj}(G)\otimes {\Lambda }^{2}V$, where $\mathrm{adj}(G)$ is the adjoint representation. In other words we derive the multiplicity with which summands of $\mathrm{adj}(G)$ appear in a tensor product $V\otimes V^{\prime }$ or (anti)symmetric square $S^{2}V$ or ${\Lambda }^{2}V$. We find in particular that the dimension of the $G$-invariant subspace of $\mathrm{adj}(G)\otimes S^{2}V$ is larger than (resp. smaller or equal to) that of $\mathrm{adj}(G)\otimes {\Lambda }^{2}V$ for a symplectic (resp. orthogonal) representation $V$.

arXiv comment14 pages. The $\mu =\overline{\nu }$ case of our Theorem 1.3, as well as our Theorem 1.4, already appear in the literature; we added the relevant references