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Introduction to Cluster Algebras. Chapter 6
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This is a preliminary draft of Chapter 6 of our forthcoming textbook "Introduction to Cluster Algebras." Chapters 1-3 have been posted as arXiv:1608.05735. Chapters 4-5 have been posted as arXiv:1707.07190. This installment contains: Chapter 6. Cluster structures in commutative rings
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Cited by 5 Pith papers
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Gr\"obner Cones for Finite Type Cluster Algebras
Compatibility-degree weights of cluster variables lie in the Gröbner cone of any finite-type cluster algebra, yielding explicit circular term orders and, for classical types, complete ray and lineality descriptions.
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Cluster structures on $SL_n/SO_n$
SL_n/SO_n, its strata ˚S_w, and the symmetric matrices Sym_n admit cluster algebra structures, obtained by folding the cluster structure on SL_n.
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Electrical networks, Grassmannians, and cluster algebras
For odd n the circular-minor cluster algebra CM_n is isomorphic to the Grassmannian cluster algebra A_{n-1,2n} after freezing/trivializing n central variables, relating circular total positivity to Grassmannian positi...
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Upper cluster structure on Kac--Moody Richardson varieties
Open Richardson varieties in symmetrizable Kac-Moody flag varieties, including twisted-product cases, are shown to carry upper cluster algebra coordinate rings.
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Cluster structures on Cox rings
A general lifting procedure produces graded upper cluster algebra structures, or the unique candidates for them, on Cox rings, with applications to flag varieties and a new diagonal partial compactification.
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