Membership in moment cones, quiver semi-invariants, and generic semi-stability for bipartite quivers
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Let Q be a bipartite quiver with vertex set Q_0 such that the number of arrows between any two source and sink vertices is constant. Let β=(β(x))_x ∈ Q_0 be a dimension vector of Q with positive integer coordinates, and let Δ(Q, β) be the moment cone associated to (Q, β). We show that the membership problem for Δ(Q, β) can be solved in strongly polynomial time. As a key step in our approach, we first solve the polytopal problem for semi-invariants of Q and its flag-extensions. Specifically, let Q_β be the flag-extension of Q obtained by attaching a flag ℱ(x) of length β(x)-1 at every vertex x of Q, and let β be the extension of β to Q_β that takes values 1, …, β(x) along the vertices of the flag ℱ(x) for every vertex x of Q. For an integral weight σ of Q_β, let K_σ be the dimension of the space of semi-invariants of weight σ on the representation space of β-dimensional complex representations of Q_β. We show that K_σ can be expressed as the number of lattice points of a certain hive-type polytope. This polytopal description together with Derksen-Weyman's Saturation Theorem for quiver semi-invariants allows us to use Tardos's algorithm to solve the membership problem for Δ(Q,β) in strongly polynomial time. In particular, this yields a strongly polynomial time algorithm for solving the generic semi-stability problem for representations of Q and Q_β.
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