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We determine the explicit transformation under duality of generic configurations of four flags inPGL   (  3  ,  C  )    in cross‐ratio coordinates. As an application, we prove the invariance under duality of an invariant in the Bloch group obtained from decorated triangulations of 3‐manifolds.

Duality and invariants of representations of fundamental groups of 3‐manifolds into PGL ( 3 , C )