Cluster Algebras and Classical Invariant Rings.

[摘要] Let V be a k-dimensional complex vector space.The Plucker 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 Plucker 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;; Plucker 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 Plucker 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.

[发布机构] University of Michigan

[效力级别] Grassmannians

[关键词] Cluster Algebras;Grassmannians;Invariant Theory;Mathematics;Science;Mathematics