Braid group symmetries of Grassmannian cluster algebras

Fraser, Chris

doi:10.1007/s00029-020-0542-3

Cited by 20 publications

(29 citation statements)

References 41 publications

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“…However, ref [32]. also pointed out at least some of these additional limiting rays are related by a braid group[107] to the limiting rays we study in this section. Therefore, even if the algebraic letters associated with these other limiting rays appear in the 8-point MHV symbol alphabet, it seems plausible they could be derived through braid transformations of the symbol alphabet derived in this section.…”

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confidence: 61%

Algebraic branch points at all loop orders from positive kinematics and wall crossing

Herderschee

2021

J. High Energ. Phys.

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There is a remarkable connection between the boundary structure of the positive kinematic region and branch points of integrated amplitudes in planar $$ \mathcal{N} $$ N = 4 SYM. A long-standing question has been precisely how algebraic branch points emerge from this picture. We use wall crossing and scattering diagrams to systematically study the boundary structure of the positive kinematic regions associated with MHV amplitudes. The notion of asymptotic chambers in the scattering diagram naturally explains the appearance of algebraic branch points. Furthermore, the scattering diagram construction also motivates a new coordinate system for kinematic space that rationalizes the relations between algebraic letters in the symbol alphabet. As a direct application, we conjecture a complete list of all algebraic letters that could appear in the symbol alphabet of the 8-point MHV amplitude.

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confidence: 61%

Algebraic branch points at all loop orders from positive kinematics and wall crossing

Herderschee

2021

J. High Energ. Phys.

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“…It is straightforward to compute the variable G = B(1 • g 3 ) = ch(T 3 ) associated to the lattice point g 3 . Remarkably we find that G, which has torus weight 2 in each of the eight Z i , is related to A by a braid element [43] of the G(4, 8) cluster modular group. Under the same braid transformation we find that B → B where B = B 2345 4567 1678 1238 .…”

Section: Jhep03(2021)065 4 G(4 8) and The Four-mass Boxmentioning

confidence: 83%

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

Arkani-Hamed

Lam

Spradlin

2021

J. High Energ. Phys.

210

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There is a remarkable well-known connection between the G(4, n) cluster algebra and n-particle amplitudes in $$ \mathcal{N} $$ N = 4 SYM theory. For n ≥ 8 two long-standing open questions have been to find a mathematically natural way to identify a finite list of amplitude symbol letters from among the infinitely many cluster variables, and to find an explanation for certain algebraic functions, such as the square roots of four-mass-box type, that are expected to appear in symbols but are not cluster variables. In this letter we use the notion of “stringy canonical forms” to construct polytopal realizations of certain compactifications of (the positive part of) the configuration space Confn(ℙk−1) ≅ G(k, n)/T that are manifestly finite for all k and n. Some facets of these polytopes are naturally associated to cluster variables, while others are naturally associated to algebraic functions constructed from Lusztig’s canonical basis. For (k, n) = (4, 8) the latter include precisely the expected square roots, revealing them to be related to certain “overpositive” functions of the kinematical invariants.

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“…Fomin and Pylyavskyy [27,Conjecture 9.3] conjectured that every cluster monomial in C[Gr (3, m)] is a web invariant. This is known in the finite mutation type cases m ≤ 9 [21]. We state the following more specific version.…”

Section: Fomin and Pylyavskyy's Conjecturesmentioning

confidence: 99%

“…An action of the extended affine braid group on Grassmannian cluster algebras was introduced in [21]. It yields an action on cluster monomials, hence on a subset of tableaux.…”

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confidence: 99%

Quantum affine algebras and Grassmannians

et al. 2020

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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.

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Braid group symmetries of Grassmannian cluster algebras

Cited by 20 publications

References 41 publications

Algebraic branch points at all loop orders from positive kinematics and wall crossing

Algebraic branch points at all loop orders from positive kinematics and wall crossing

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

Quantum affine algebras and Grassmannians

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