Cluster algebras and classical invariant rings
نویسنده
چکیده
Let V be a k-dimensional complex vector space. The Plücker ring of polynomial SL(V ) invariants of a collection of n vectors in V can be alternatively described as the homogeneous coordinate ring of the Grassmannian Gr(k, n). In 2003, using combinatorial tools developed by A. Postnikov, J. Scott showed that the Plücker ring carries a cluster algebra structure. Over the ensuing decade, this has become one of the central examples of cluster algebra theory. In the 1930s, H. Weyl described the structure of the “mixed” Plücker ring, the ring of polynomial SL(V ) invariants of a collection of n vectors in V and m covectors in V ∗. In this thesis, we generalize Scott’s construction and Postnikov’s combinatorics to this more general setting. In particular, we show that each mixed Plücker ring carries a natural cluster algebra structure, which was previously established by S. Fomin and P. Pylyavskyy only in the case k = 3. We also introduce mixed weak separation as a combinatorial condition for compatibility of cluster variables in this cluster structure and prove that maximal collections of weakly separated mixed subsets satisfy a purity result, a property proved in the Grassmannian case by Oh, Postnikov, and Speyer.
منابع مشابه
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