Determine the algebra of invariant valuations for real Grassmannians

Determine the algebra of translation-invariant valuations on the tensor space \(\mathbb{R}^k\otimes\mathbb{R}^m\) that are invariant under the action of \(O(k)\times O(m)\), thereby characterizing the probabilistic intersection ring of the real Grassmannian \(G(k,m)\).

Background

For real Grassmannians G(k,m)G(k,m) with k,m>1k,m>1, the isotropy action of O(k)×O(m)O(k)\times O(m) on Rk⊗Rm\mathbb{R}^k\otimes\mathbb{R}^m is not transitive on the unit sphere. Consequently, the associated probabilistic intersection ring is infinite-dimensional and substantially more difficult to analyze than the finite-dimensional examples of spheres and complex projective spaces.

The paper develops Schubert zonoids and proves an exterior-power decomposition into Schubert spans, but these results do not determine the full multiplication structure of the invariant ring. The authors explicitly identify its complete algebraic description as unresolved.

References

Despite this, the true nature of the probabilistic intersection ring of real Grassmannian, still remains to be discovered. In the language of valuations, this amounts to determine the algebra of translation invariant valuations on Rk \otimes Rm that are invariant under the action of the group O(k) \times O(m).

— Probabilistic intersection theory in Riemannian homogeneous spaces  (2502.08256 - Breiding et al., 12 Feb 2025) in Section 1, subsection “Probabilistic Schubert Calculus”