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We study sets of commuting reflection functors in the derived category ofsheaves on Calabi-Yau varieties. We show that such a collection is determinedby a set of mutually orthogonal spherical objects. We also show that when thespherical objects are locally-free sheaves then the kernel of the compositetransform parametrizes properly torsion-free with zero-dimensional singularitysets and conversely that such a kernel gives rise to a collection of mutuallyorthogonal spherical vector bundles. We do this using a more detailed analysisof the reason why spherical twists give equivalences.

A Note on Commuting Reflection Functors for Calabi-Yau d-folds