Tensor invariants for classical groups revisited

We reconsider an old problem, namely the dimension of the $G$-invariant subspace in $V^{\otimes p}\otimes V^{\ast \otimes q}$, where $G$ is one of the classical groups $\mathrm{G}\mathrm{L}(V)$, $\mathrm{S}\mathrm{L}(V)$, $\mathrm{O}(V)$, $\mathrm{S}\mathrm{O}(V)$, or $\mathrm{S}\mathrm{p}(V)$. Spanning sets for the invariant subspace have long been well known, but linear bases are more delicate. The main contribution of this paper is a combinatorial realization of linear bases via standard Young tableaux and arc diagrams, in a uniform manner for all five classical groups. As a secondary contribution, we survey the many equivalent ways – some old, some new – to enumerate the elements in these bases.