Quantum affine algebras and Grassmannians

Abstract

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.

Non-perturbative geometries for planar $$ \mathcal{N} $$ = 4 SYM amplitudes

“…+ . In the following it will be important that we always use a cluster parameterization, which means that the parameters are Fock-Goncharov coordinates in the initial cluster of the G(k, n) cluster algebra (for which we use the conventions of figure 2 of [27]). For example,…”

“…Specifically, in the infinite case (4, 8) we find that the normal rays to 356 of the facets of C(4, 8) lie along g-vectors of the G(4, 8) cluster algebra, consistent with the results reported in [33,37,38] for the fan associated to Trop + G(4, 8). However, the remaining 4 normal rays are not even inside the cluster fan, according to the criterion given in Theorem 3.25 of [27]. The normal rays to the 274 facets of C † (4, 8) are a proper subset of those of the facets of C(4, 8); 272 of them lie along g-vectors and 2 of them are outside the cluster fan.…”

“…There is a large literature on the construction of various "canonical bases" of cluster algebras in infinite type. For G(k, n) every integer point in g-vector space can be assigned [27] (partly conjecturally) a polynomial in cluster variables, Lusztig's canonical basis element [40]. When the integer point lies inside a g-vector cone of some cluster, the corresponding function is simply the obvious associated monomial in cluster variables, but when the integer point is on a ray not pointing in the direction of a cluster variable, the corresponding basis element is not a cluster variable, but something else.…”

“…The first two of these are also normal rays to C † (4,8). We adopt Corollary 7.3 of [27] as a (conjectural) way to assign a canonical basis element to any lattice point in R d , regardless of whether it is in the cluster fan. In the notation defined in that paper, the semistandard Young tableaux T 1 , .…”

“…. , T 4 associated to (4.1) are respectively Let us first consider T 1 , which is the same as the T 4 considered in Example 8.1 of [27], where the variables associated to the first two points along the ray generated by g 1 were computed as We have computed the next variable along this ray,…”