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We provide a concrete realization of the cluster algebras associated with   $Q$  -systems as amalgamations of cluster structures on double Bruhat cells in simple algebraic groups. For nonsimply laced groups, this provides a cluster-algebraic formulation of   $Q$  -systems of twisted type. It also yields a uniform proof of the discrete integrability of   $Q$  -systems of different types by identifying them with the dynamics of factorization mappings on quotients of double Bruhat cells. On the double Bruhat cell itself, we find these dynamics are closely related to those of the Fomin–Zelevinsky twist map. This leads to an explicit formula expressing twisted cluster variables as Laurent monomials in the untwisted cluster variables obtained from the corresponding mutation sequence. This holds for Coxeter double Bruhat cells in any symmetrizable Kac–Moody group, and in affine type the factorization mapping is again an integrable mutation sequence.

Q-Systems, Factorization Dynamics, and the Twist Automorphism