Solution of tetrahedron equation and cluster algebras

Abstract We notice a remarkable connection between the Bazhanov-Sergeev solution of Zamolodchikov tetrahedron equation and certain well-known cluster algebra expression. The tetrahedron transformation is then identified with a sequence of four mutations. As an application of the new formalism, we show how to construct an integrable system with the spectral curve with arbitrary symmetric Newton polygon. Finally, we embed this integrable system into the double Bruhat cell of a Poisson-Lie group, show how triangular decomposition can be used to extend our approach to the general non-symmetric Newton polygons, and prove the Lemma which classifies conjugacy classes in double affine Weyl groups of A-type by decorated Newton polygons.

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CLUSTER ALGEBRAS FROM SURFACES AND EXTENDED AFFINE WEYL GROUPS

Transformation Groups ◽

10.1007/s00031-021-09647-y ◽

2021 ◽

Author(s):

ANNA FELIKSON ◽

MICHAEL SHAPIRO ◽

JOHN W. LAWSON ◽

PAVEL TUMARKIN

Keyword(s):

Weyl Group ◽

Quadratic Forms ◽

Cluster Algebras ◽

Quadratic Space ◽

Weyl Groups ◽

Affine Weyl Groups ◽

Affine Weyl Group

AbstractWe characterize mutation-finite cluster algebras of rank at least 3 using positive semi-definite quadratic forms. In particular, we associate with every unpunctured bordered surface a positive semi-definite quadratic space V , and with every triangulation a basis in V , such that any mutation of a cluster (i.e., a flip of a triangulation) transforms the corresponding bases into each other by partial reflections. Furthermore, every triangulation gives rise to an extended affine Weyl group of type A, which is invariant under flips. The construction is also extended to exceptional skew-symmetric mutation-finite cluster algebras of types E.

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Correction to: CLUSTER ALGEBRAS FROM SURFACES AND EXTENDED AFFINE WEYL GROUPS

Transformation Groups ◽

10.1007/s00031-021-09663-y ◽

2021 ◽

Author(s):

ANNA FELIKSON ◽

JOHN W. LAWSON ◽

MICHAEL SHAPIRO ◽

PAVEL TUMARKIN

Keyword(s):

Cluster Algebras ◽

Weyl Groups ◽

Affine Weyl Groups

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Notes on cluster algebras and some all-loop Feynman integrals

Journal of High Energy Physics ◽

10.1007/jhep06(2021)119 ◽

2021 ◽

Vol 2021 (6) ◽

Author(s):

Song He ◽

Zhenjie Li ◽

Qinglin Yang

Keyword(s):

Configuration Space ◽

Strong Evidence ◽

Wilson Loop ◽

Cluster Algebra ◽

Cluster Algebras ◽

Gauge Invariant ◽

Feynman Integrals ◽

Cluster Function ◽

Invariant Description ◽

Cluster Configuration

Abstract We study cluster algebras for some all-loop Feynman integrals, including box-ladder, penta-box-ladder, and double-penta-ladder integrals. In addition to the well-known box ladder whose symbol alphabet is $$ {D}_2\simeq {A}_1^2 $$ D 2 ≃ A 1 2 , we show that penta-box ladder has an alphabet of D3 ≃ A3 and provide strong evidence that the alphabet of seven-point double-penta ladders can be identified with a D4 cluster algebra. We relate the symbol letters to the u variables of cluster configuration space, which provide a gauge-invariant description of the cluster algebra, and we find various sub-algebras associated with limits of the integrals. We comment on constraints similar to extended-Steinmann relations or cluster adjacency conditions on cluster function spaces. Our study of the symbol and alphabet is based on the recently proposed Wilson-loop d log representation, which allows us to predict higher-loop alphabet recursively; by applying it to certain eight-point and nine-point double-penta ladders, we also find D5 and D6 cluster functions respectively.

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On Generalized Minors and Quiver Representations

International Mathematics Research Notices ◽

10.1093/imrn/rny053 ◽

2018 ◽

Vol 2020 (3) ◽

pp. 914-956 ◽

Cited By ~ 1

Author(s):

Dylan Rupel ◽

Salvatore Stella ◽

Harold Williams

Keyword(s):

Finite Type ◽

Dynkin Diagram ◽

Cluster Algebra ◽

Coordinate Ring ◽

Coxeter Element ◽

Quiver Representations ◽

Group Representations ◽

Weight Group ◽

Highest Weight ◽

Bruhat Cell

Abstract The cluster algebra of any acyclic quiver can be realized as the coordinate ring of a subvariety of a Kac–Moody group—the quiver is an orientation of its Dynkin diagram, defining a Coxeter element and thereby a double Bruhat cell. We use this realization to connect representations of the quiver with those of the group. We show that cluster variables of preprojective (resp. postinjective) quiver representations are realized by generalized minors of highest-weight (resp. lowest-weight) group representations, generalizing results of Yang–Zelevinsky in finite type. In type $A_{n}^{\!(1)}$ and finitely many other affine types, we show that cluster variables of regular quiver representations are realized by generalized minors of group representations that are neither highest- nor lowest-weight; we conjecture this holds more generally.

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Bijective projections on parabolic quotients of affine Weyl groups

Journal of Algebraic Combinatorics ◽

10.1007/s10801-014-0559-9 ◽

2014 ◽

Vol 41 (4) ◽

pp. 911-948 ◽

Cited By ~ 1

Author(s):

Elizabeth Beazley ◽

Margaret Nichols ◽

Min Hae Park ◽

XiaoLin Shi ◽

Alexander Youcis

Keyword(s):

Weyl Groups ◽

Affine Weyl Groups

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Tight Quotients and Double Quotients in the Bruhat Order

The Electronic Journal of Combinatorics ◽

10.37236/1871 ◽

2005 ◽

Vol 11 (2) ◽

Cited By ~ 5

Author(s):

John R. Stembridge

Keyword(s):

Coxeter Groups ◽

Upper Bounds ◽

Bruhat Order ◽

Weyl Groups ◽

Order Dimension ◽

Parabolic Subgroups ◽

Affine Weyl Groups ◽

Positive Side ◽

Maximal Parabolic Subgroups

It is a well-known theorem of Deodhar that the Bruhat ordering of a Coxeter group is the conjunction of its projections onto quotients by maximal parabolic subgroups. Similarly, the Bruhat order is also the conjunction of a larger number of simpler quotients obtained by projecting onto two-sided (i.e., "double") quotients by pairs of maximal parabolic subgroups. Each one-sided quotient may be represented as an orbit in the reflection representation, and each double quotient corresponds to the portion of an orbit on the positive side of certain hyperplanes. In some cases, these orbit representations are "tight" in the sense that the root system induces an ordering on the orbit that yields effective coordinates for the Bruhat order, and hence also provides upper bounds for the order dimension. In this paper, we (1) provide a general characterization of tightness for one-sided quotients, (2) classify all tight one-sided quotients of finite Coxeter groups, and (3) classify all tight double quotients of affine Weyl groups.

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New Combinatorial Formulas for Cluster Monomials of Type $A$ Quivers

The Electronic Journal of Combinatorics ◽

10.37236/6464 ◽

2017 ◽

Vol 24 (2) ◽

Author(s):

Kyungyong Lee ◽

Li Li ◽

Ba Nguyen

Keyword(s):

Cluster Algebra ◽

Theta Functions ◽

Type A ◽

Perfect Matchings ◽

Cluster Algebras ◽

Combinatorial Formula ◽

Natural Basis ◽

Combinatorial Formulas ◽

New Formula

Lots of research focuses on the combinatorics behind various bases of cluster algebras. This paper studies the natural basis of a type $A$ cluster algebra, which consists of all cluster monomials. We introduce a new kind of combinatorial formula for the cluster monomials in terms of the so-called globally compatible collections. We give bijective proofs of these formulas by comparing with the well-known combinatorial models of the $T$-paths and of the perfect matchings in a snake diagram. For cluster variables of a type $A$ cluster algebra, we give a bijection that relates our new formula with the theta functions constructed by Gross, Hacking, Keel and Kontsevich.

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