Abstract. We introduce quasi-homomorphisms of cluster algebras, a flexible notion of a map between cluster algebras of the same type (but with different coefficients). The definition is given in terms of seed orbits, the smallest equivalence classes of seeds on which the mutation rules for non-normalized seeds are unambiguous. We present examples of quasi-homomorphisms involving familiar cluster algebras, such as cluster structures on Grassmannians, and those associated with marked surfaces with boundary. We explore the related notion of a quasi-automorphism, and compare the resulting group with other groups of symmetries of cluster structures. For cluster algebras from surfaces, we determine the subgroup of quasi-automorphisms inside the tagged mapping class group of the surface.
We study the relation between quantum affine algebras of type A and Grassmannian cluster algebras. Hernandez and Leclerc described an isomorphism from the Grothendieck ring of a certain subcategory C ℓ of U q ( sl n )-modules to a quotient of the Grassmannian cluster algebra in which certain frozen variables are set to 1. We explain how this induces an isomorphism between the monoid of dominant monomials, used to parameterize simple modules, and a quotient of the monoid of rectangular semistandard Young tableaux with n rows and with entries in [n + ℓ + 1]. Via the isomorphism, we define an element ch(T ) in a Grassmannian cluster algebra for every rectangular tableau T . By results of Kashiwara, Kim, Oh, and Park, and also of Qin, every Grassmannian cluster monomial is of the form ch(T ) for some T . Using a formula of Arakawa-Suzuki, we give an explicit expression for ch(T ), and also give explicit q-character formulas for finite-dimensional U q ( sl n )-modules. We give a tableau-theoretic rule for performing mutations in Grassmannian cluster algebras. We suggest how our formulas might be used to study reality and primeness of modules, and compatibility of cluster variables.
We formulate a higher-rank version of the boundary measurement map for weighted planar bipartite networks in the disk. It sends a network to a linear combination of SL r -webs, and is built upon the r-fold dimer model on the network. When r equals 1, our map is a reformulation of Postnikov's boundary measurement used to coordinatize positroid strata. When r equals 2 or 3, it is a reformulation of the SL 2 -and SL 3 -web immanants defined by the second author. The basic result is that the higher rank map factors through Postnikov's map. As an application, we deduce generators and relations for the space of SL r -webs, reproving a result of Cautis-Kamnitzer-Morrison. We establish compatibility between our map and restriction to positroid strata, and thus between webs and total positivity.
Let Gr • (k, n) ⊂ Gr(k, n) denote the open positroid stratum in the Grassmannian. We define an action of the extended affine d-strand braid group on Gr • (k, n) by regular automorphisms, for d the greatest common divisor of k and n. The action is by quasi-automorphisms of the cluster structure on Gr • (k, n), determining a homomorphism from the extended affine braid group to the cluster modular group for Gr(k, n). We also define a quasi-isomorphism between the Grassmannian Gr(k, rk) and the Fock-Goncharov configuration space of 2r-tuples of affine flags for SL k . This identifies the cluster variables, clusters, and cluster modular groups, in these two cluster structures.Fomin and Pylyavskyy proposed a description of the cluster combinatorics for Gr(3, n) in terms of Kuperberg's basis of non-elliptic webs. As our main application, we prove many of their conjectures for Gr (3,9) and give a presentation for its cluster modular group. We establish similar results for Gr(4, 8). These results rely on the fact that both of these Grassmannians have finite mutation type.2010 Mathematics Subject Classification. 13F60.
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