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I've seen hints at the following result:

Let $M$ be a 3-dimensional manifold and let $T := T^*M$ be the cotangent bundle. By Schur-Weyl Duality, the 3rd tensor product can be written as follows: $$T^{\otimes 3} = \mathrm{Sym}^3(T) \oplus \mathrm{Alt}^3(T) \oplus S_{\lambda}(T^{\otimes 3})$$ where $\lambda$ is the partition $3=1+2$. What geometric information does $S_{\lambda}$ give? I've heard reference that certain curvature tensors may live in this bundle, though I'm not sure.

For a smooth manifold of dimension $n$, what geometric results are there on the various Schur Functions $S_{\lambda}(T^{\otimes n})$?

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    $\begingroup$ I don't know a geometric interpretation, but you need two copies of that rep. $\endgroup$
    – Phil Tosteson
    Commented Oct 1, 2019 at 2:43
  • $\begingroup$ Which representation? Why do I need two? $\endgroup$
    – Rdrr
    Commented Oct 1, 2019 at 3:25

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