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This paper shows that the motivic BPS invariants associated to anoncommutative crepant resolution of a compound Du-Val singularity arecontrolled by the labelled Dynkin combinatorics appearing in the work ofIyama--Wemyss. In particular, we show that the invariants vanish for dimensionvectors which are not a multiple of a restricted root obtained from the affineroot system under a natural quotient map. An immediate corollary is adescription of the curve classes for which the Gopakumar--Vafa invariants ofgeometric crepant resolutions vanish, generalising a recent result ofNabijou--Wemyss to the nonisolated setting. We furthermore formulate a methodfor finding symmetries among the non-vanishing invariants using derivedequivalences, and show how this can be applied to line bundle twists andmutation functors in some settings. In particular, we find new wall-crossingrelations among the Gopakumar-Vafa invariants of the different crepantresolutions of a cDV singularity.

Vanishing and Symmetries of BPS Invariants for CDV Singularities