Tensor invariants for classical groups revisited

Abstract

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 $\mathrm{GL}(V)$, $\mathrm{SL}(V)$, $\mathrm{O}(V)$, $\mathrm{SO}(V)$, or $\mathrm{Sp}(V)$. 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.

25 pages + appendix; version 2 modifies the structure and exposition of the paper by presenting all preliminary results in Sections 2 and 3