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We study the coinduction functor on the category of FI-modules and its variants. Using the coinduction functor, we give new and simpler proofs of (generalizations of) various results on homological properties of FI-modules. We also prove that any finitely generated projective VI-module over a field of characteristic 0 is injective.

It is known that many (upper) cluster algebras possess different kinds of good bases which contain the cluster monomials and are parametrized by the tropical points of cluster Poisson varieties. For a large class of upper cluster algebras (injective-reachable ones with full rank coefficients), we describe all of its bases with these properties. Moreover, we show the existence of the generic basis for them. In addition, we prove that Bridgeland's representation theoretic formula is effective for their theta functions (weak genteelness). Our results apply to (almost) all well-known cluster algebras arising from representation theory or higher Teichmüller theory, including quantum affine algebras, unipotent cells, double Bruhat cells, skein algebras over surfaces, where we change the coefficients if necessary so that the full rank assumption holds.

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We study the oriented exchange graph EG(Gamma_N Q) ofreachable hearts in the finite-dimensional derived category D(Gamma_N Q) of the CY-N Ginzburg algebra Gamma_N Q associated to an acyclic quiver Q. We show that any such heart is induced from some heart in the bounded derived category D(Q) via some ‘Lagrangian immersion’ L :D(Q)->D(Gamma_N Q). We build on this to show that the quotient of EG(Gamma_N Q) by the Seidel–Thomas braid group is the exchange graph CEG_N−1(Q)of cluster tilting sets in the (higher) cluster category C_N−1(Q). As an application, we interpret Buan–Thomas’ coloured quiver for a cluster tilting set in terms of the Ext quiver of any corresponding heart in D(Gamma_N Q).