For crepant resolutions X(1,3,9) and X(1,3,13) the derived categories admit faithful braid twist group actions of types D and E induced by spherical object configurations.
Craw, An explicit construction of the McKay correspondence for A -Hilb ( C ^3) , J
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The derived category D(X) of the crepant resolution X of the 1/7(1,2,4) singularity admits a faithful action by the braid group associated to a 3-cycle quiver.
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Some faithful algebraic braid twist group actions for 3-fold crepant resolutions
For crepant resolutions X(1,3,9) and X(1,3,13) the derived categories admit faithful braid twist group actions of types D and E induced by spherical object configurations.