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Tensor diagrams and cluster algebras. (English) Zbl 1386.13062

A cluster algebra, as invented by S. Fomin and A. Zelevinsky [J. Am. Math. Soc. 15, No. 2, 497–529 (2002; Zbl 1021.16017)], is a commutative algebra generated by a family of generators called cluster variables. Homogeneous coordinate rings of Grassmannians are among the most important examples of cluster algebras and provide the most concrete and accessible examples of naturally defined cluster algebras of infinite mutation type. Cluster structures in these rings play a prominent role in applications of cluster theory arising in connection with other mathematical fields. So far apart from a few special cases, cluster structures on Grassmannians are not well understood.
Let \(\text{Gr}_{k,N}\) denote the Grassmann manifold of \(k\)-subspaces in an \(N\)-dimensional complex vector space. The corresponding cluster algebra has finite type if and only if \((k-2)(N-k-2)\leq 3\). The authors advocated the point of view that many aspects of cluster structures on Grassmannians are best understood within a broader range of examples coming from classical invariant theory. Recall that the homogeneous coordinate ring of \(\text{Gr}_{k,N}\) is isomorphic to the ring of \(\text{SL}(V)\) invariants of \(N\)-tuples of vectors in a \(k\)-dimensional complex vector space \(V\), which were explicitly described by Hermann Weyl in the 1930s. More general rings of \(\text{SL}(V)\) invariants of collections of vectors and linear forms have been thoroughly studied by classical invariant theory. The authors conjectured that every such ring carries a natural cluster algebra structure.
In the paper under review, the authors showed that when \(V\) is \(3\)-dimensional, each of these rings carries a natural cluster algebra structure (typically, many of them) whose cluster variables include Weyl’s generators. They described and explored these cluster structures using the combinatorial machinery of tensor diagrams. The distinguishing characteristic of the \(3\)-dimensional case is the existence of a web basis discovered by G. Kuperberg [Commun. Math. Phys. 180, No. 1, 109–151 (1996; Zbl 0870.17005)]. The authors’ approach was an emphasis on the multiplicative properties of the web basis, which along with its compatibility with tensor contraction play a central role in the study of cluster algebra structures in these rings of invariants.

MSC:

13F60 Cluster algebras
05E99 Algebraic combinatorics
13A50 Actions of groups on commutative rings; invariant theory
15A72 Vector and tensor algebra, theory of invariants
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