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Quivers, cones and polytopes. (English) Zbl 1034.52011
Summary: Let $$Q$$ be a quiver without oriented cycles. We consider the polytope of flows $$\Delta(\theta)$$ in $$Q$$ with input $$\theta$$. These polytopes are closely related to the combinatorial structure of the quiver, in particular, to its spanning subtrees. Furthermore, we consider a system of cones which turns out to be a fan and can be seen as a base for the family of all flow polytopes $$\Delta(\theta)$$ for the various inputs $$\theta$$. Finally, we present several examples.

MSC:
 52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.) 14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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References:
 [1] Altmann, K.; Hille, L., Strong exceptional sequences provided by quivers, Algebraicrepresent.theory, 2, 1, 1-17, (1999) · Zbl 0951.16006 [2] Batyrev, V.V., Dual polyhedra and mirror symmetry for calabi – yau hypersurfaces in toric varieties, J. algebraic geom., 3, 3, 493-535, (1994) · Zbl 0829.14023 [3] Batyrev, V.V.; Ciocan-Fontanine, I.; Kim, B.; van Straten, D., Mirror symmetry and toric degenerations of partial flag manifolds, Acta math., 184, 1, 1-39, (2000) · Zbl 1022.14014 [4] Fulton, W., Introduction to toric varieties, () · Zbl 1083.14065 [5] Hille, L., Toric quiver varieties, algebras and modules, II (geiranger, 1996), (), 311-325 [6] G. Kempf, F.F. Knudsen, D. Mumford, B. Saint-Donat, Toroidal Embeddings. I. Lecture Notes in Mathematics, vol. 339, Springer, Berlin, 1973 · Zbl 0271.14017 [7] King, A.D., Moduli of representations of finite dimensional algebras, Quart. J. math. Oxford, 45, 2, 515-530, (1994) · Zbl 0837.16005 [8] T. Oda, Convex bodies and algebraic geometry. An introduction to the theory of toric varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete, 15 (3) (1988) · Zbl 0628.52002 [9] M. Reid, What is a Flip, Preprint, Colloquium Talk at Utah [10] Thaddeus, M., Geometric invariant theory and flips, J. amer. math. soc., 9, 3, 691-723, (1996) · Zbl 0874.14042
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