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Grassmannians and cluster algebras. (English) Zbl 1088.22009

This paper studies and clarifies the interdependence between cluster algebra primer, Postnikov arrangements, quadrilateral arrangement, cluster algebra of geometric type, toric charts and positivity, Grassmannians of finite type, Laurent positivity and Schur polynomials.
Main result: The only Grassmannians \(G(k,n)\), within the range of indices \(2<k\leq {1\over 2}n\), whose homogeneous coordinate rings are of finite type, are the Grassmannians \(G(3,6)\), \(G(3,7)\) and \(G(3,8)\). As cluster algebras, their coordinate rings correspond, respectively, to the root systems \(D_4\), \(E_6\) and \(E_8\). The pairing between cluster variables and almost positive roots is explicitly worked out in each case and a geometric description of all cluster variables is presented.

MSC:

22E46 Semisimple Lie groups and their representations
05E99 Algebraic combinatorics
14M15 Grassmannians, Schubert varieties, flag manifolds