scholarly journals Solutions to the T-Systems with Principal Coefficients

10.37236/5698 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Panupong Vichitkunakorn
Keyword(s):  
Recurrence Relation ◽  
Cluster Algebra ◽  
Perfect Matchings ◽  
Partition Functions ◽  
Initial Condition ◽  
Free Cluster ◽  
Octahedron Recurrence ◽  
T System

The $A_\infty$ T-system, also called the octahedron recurrence, is a dynamical recurrence relation. It can be realized as mutation in a coefficient-free cluster algebra (Kedem 2008, Di Francesco and Kedem 2009). We define T-systems with principal coefficients from cluster algebra aspect, and give combinatorial solutions with respect to any valid initial condition in terms of partition functions of perfect matchings, non-intersecting paths and networks. This also provides a solution to other systems with various choices of coefficients on T-systems including Speyer's octahedron recurrence (Speyer 2007), generalized lambda-determinants (Di Francesco 2013) and (higher) pentagram maps (Schwartz 1992, Ovsienko et al. 2010, Glick 2011, Gekhtman et al. 2016).

scholarly journals Positivity of the T-System Cluster Algebra

10.37236/229 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Philippe Di Francesco ◽  
Rinat Kedem
Keyword(s):  
Continued Fraction ◽  
Cluster Algebra ◽  
Path Model ◽  
Partition Functions ◽  
Weighted Graphs ◽  
Generic Boundary ◽  
Seed Data ◽  
Q System ◽  
Extra Parameter ◽  
T System

We give the path model solution for the cluster algebra variables of the $T$-system of type $A_r$ with generic boundary conditions. The solutions are partition functions of (strongly) non-intersecting paths on weighted graphs. The graphs are the same as those constructed for the $Q$-system in our earlier work, and depend on the seed or initial data in terms of which the solutions are given. The weights are "time-dependent" where "time" is the extra parameter which distinguishes the $T$-system from the $Q$-system, usually identified as the spectral parameter in the context of representation theory. The path model is alternatively described on a graph with non-commutative weights, and cluster mutations are interpreted as non-commutative continued fraction rearrangements. As a consequence, the solution is a positive Laurent polynomial of the seed data.

scholarly journals BPS Quivers of Five-Dimensional SCFTs, Topological Strings and q-Painlevé Equations

2021 ◽  
Author(s):  
Giulio Bonelli ◽  
Fabrizio Del Monte ◽  
Alessandro Tanzini
Keyword(s):  
Topological String ◽  
Cluster Algebra ◽  
Painlevé Equations ◽  
Partition Functions ◽  
Quantum Field Theories ◽  
Particle Spectrum ◽  
Yang Mills ◽  
Painleve Equations ◽  
Rank One ◽  
Bilinear Equations

AbstractWe study the discrete flows generated by the symmetry group of the BPS quivers for Calabi–Yau geometries describing five-dimensional superconformal quantum field theories on a circle. These flows naturally describe the BPS particle spectrum of such theories and at the same time generate bilinear equations of q-difference type which, in the rank one case, are q-Painlevé equations. The solutions of these equations are shown to be given by grand canonical topological string partition functions which we identify with $$\tau $$ τ -functions of the cluster algebra associated to the quiver. We exemplify our construction in the case corresponding to five-dimensional SU(2) pure super Yang–Mills and $$N_f=2$$ N f = 2 on a circle.

scholarly journals The Cube Recurrence

10.37236/1826 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Gabriel D. Carroll ◽  
David Speyer
Keyword(s):  
Recurrence Relation ◽  
Planar Graphs ◽  
Perfect Matchings ◽  
Laurent Polynomials ◽  
Combinatorial Interpretation ◽  
Combinatorial Model

We construct a combinatorial model that is described by the cube recurrence, a quadratic recurrence relation introduced by Propp, which generates families of Laurent polynomials indexed by points in ${\Bbb Z}^3$. In the process, we prove several conjectures of Propp and of Fomin and Zelevinsky about the structure of these polynomials, and we obtain a combinatorial interpretation for the terms of Gale-Robinson sequences, including the Somos-6 and Somos-7 sequences. We also indicate how the model might be used to obtain some interesting results about perfect matchings of certain bipartite planar graphs.

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

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.

scholarly journals $(q,t)$-Characters of Kirillov-Reshetikhin Modules of Type $A_r$ as Quantum Cluster Variables

10.37236/7188 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Bolor Turmunkh
Keyword(s):  
Dynamical System ◽  
Discrete Dynamical System ◽  
Cluster Algebra ◽  
Quiver Varieties ◽  
Quantum Cluster ◽  
T System

Nakajima (2003) introduced a $t$-deformation of $q$-characters, $(q,t)$-characters for short, and their twisted multiplication through the geometry of quiver varieties. The Nakajima $(q,t)$-characters of Kirillov-Reshetikhin modules satisfy a $t$-deformed $T$-system. The $T$-system is a discrete dynamical system that can be interpreted as a mutation relation in a cluster algebra in two different ways, depending on the choice of direction of evolution. In this paper, we show that the Nakajima $t$-deformed $T$-system of type $A_r$ forms a quantum mutation relation in a quantization of exactly one of the cluster algebra structures attached to the $T$-system.

scholarly journals The proof of the Perepechko’s conjecture concerning near-perfect matchings on Cm x Pn cylinders of odd order

2019 ◽  
Vol 13 (2) ◽  
pp. 361-377
Author(s):  
Rade Doroslovacki ◽  
Jelena Djokic ◽  
Bojana Pantic ◽  
Olga Bodroza-Pantic
Keyword(s):  
Recurrence Relation ◽  
Generating Function ◽  
Perfect Matchings ◽  
Odd Order

For all odd values of m, we prove that the sequence of the numbers of near-perfect matchings on Cm x P2n+1 cylinder with a vacancy on the boundary obeys the same recurrence relation as the sequence of the numbers of perfect matchings on Cm x P2n. Further more, we prove that for all odd values of m denominator of the generating function for the total number of the near-perfect matchings on Cm x P2n+1 graph is always the square of denominator of generating function for the sequence of the numbers of perfect matchings on Cm x P2n graph, as recently conjectured by Perepechko.

scholarly journals Periodicities of T-systems and Y-systems

2010 ◽  
Vol 197 ◽  
pp. 59-174 ◽  
Author(s):  
Rei Inoue ◽  
Osamu Iyama ◽  
Atsuo Kuniba ◽  
Tomoki Nakanishi ◽  
Junji Suzuki
Keyword(s):  
Lie Algebra ◽  
Direct Method ◽  
Cluster Algebra ◽  
Quantum Affine Algebra ◽  
Cluster Category ◽  
Affine Algebra ◽  
Grothendieck Ring ◽  
Finite Dimensional ◽  
Level 2 ◽  
T System

The unrestricted T-system is a family of relations in the Grothendieck ring of the category of the finite-dimensional modules of Yangian or quantum affine algebra associated with a complex simple Lie algebra. The unrestricted T-system admits a reduction called the restricted T-system. In this paper we formulate the periodicity conjecture for the restricted T-systems, which is the counterpart of the known and partially proved periodicity conjecture for the restricted Y-systems. Then, we partially prove the conjecture by various methods: the cluster algebra and cluster category method for the simply laced case, the determinant method for types A and C, and the direct method for types A, D, and B (level 2).

scholarly journals A Positive Proof of the Littlewood-Richardson Rule using the Octahedron Recurrence

10.37236/1814 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Allen Knutson ◽  
Terence Tao ◽  
Christopher Woodward
Keyword(s):  
Closed Form ◽  
Tensor Products ◽  
Perfect Matchings ◽  
Representation Ring ◽  
Dominant Weights ◽  
Structure Constants ◽  
Octahedron Recurrence

We define the hive ring, which has a basis indexed by dominant weights for $GL_n({\Bbb C})$, and structure constants given by counting hives [Knutson-Tao, "The honeycomb model of $GL_n$ tensor products"] (or equivalently honeycombs, or BZ patterns [Berenstein-Zelevinsky, "Involutions on Gel$'$fand-Tsetlin schemes$\dots$ "]). We use the octahedron rule from [Robbins-Rumsey, "Determinants$\dots$"] to prove bijectively that this "ring" is indeed associative. This, and the Pieri rule, give a self-contained proof that the hive ring is isomorphic as a ring-with-basis to the representation ring of $GL_n({\Bbb C})$. In the honeycomb interpretation, the octahedron rule becomes "scattering" of the honeycombs. This recovers some of the "crosses and wrenches" diagrams from Speyer's very recent preprint ["Perfect matchings$\dots$"], whose results we use to give a closed form for the associativity bijection.

scholarly journals Perfect matchings and the octahedron recurrence

2006 ◽  
Vol 25 (3) ◽  
pp. 309-348 ◽  
Author(s):  
David E. Speyer
Keyword(s):  
Perfect Matchings ◽  
Octahedron Recurrence

scholarly journals Generating functions for vector partition functions and a basic recurrence relation

2019 ◽  
Vol 25 (7) ◽  
pp. 1052-1061 ◽  
Author(s):  
Alexander P. Lyapin ◽  
Sreelatha Chandragiri
Keyword(s):  
Recurrence Relation ◽  
Generating Functions ◽  
Partition Functions
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