Killing tensors as irreducible representations of the general linear group (Q1763505): Difference between revisions

Let \((M^n, g_{\alpha,\beta})\) be an \(n\)-dimensional pseudo-Riemannian manifold of constant curvature. A Killing tensor of valence \(p\) is a symmetric tensor field \(h_{\alpha_1\dots\alpha_p}\) satisfying \(\nabla_{(\alpha_0}h_{\alpha_1\dots\alpha_p)}=0\). Let \({\mathcal K}^p\) denote the vector space of such tensor fields on \(M^n\). \textit{M.~Takeuchi} showed [Tsukuba J. Math. 7, 233--255 (1983; Zbl 0567.53017)] that \({\mathcal K}^p\) is isomorphic to a certain representation of the general linear group by invoking the Bott-Borel-Weil theorem. The authors construct an elementary isomorphism between \({\mathcal K}^p\) and this irreducible representation. This isomorphism is equivariant in the sense that the natural action of the isometry group corresponds to the restriction of the linear action to the appropriate subgroup.