## The cube recurrence

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Citations: | 21 - 0 self |

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@MISC{Carroll_thecube,

author = {Gabriel D. Carroll and David Speyer},

title = {The cube recurrence},

year = {}

}

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### Abstract

Keywords: cube recurrence, grove, Gale-Robinson theorem

### Citations

72 | Cluster algebras I: Foundations - Fomin, Zelevinsky |

70 | Moduli spaces of local systems and higher Teichmüller theory - Fock, Goncharov |

69 |
Alternating-sign matrices and domino tilings
- Elkies, Kuperberg, et al.
- 1992
(Show Context)
Citation Context ...ions with the Hirota equation in physics, with Dodgson's condensation method of evaluating determinants, with alternating-sign matrices, and with domino tilings of Aztec diamonds (see [11], [8], [9], =-=[3]-=-, the electronic journal of combinatorics 11 (2004), #R73 1srespectively). It turns out that every fi,j,k is a Laurent polynomial in the initial variables xi,j,k, i.e. a polynomial in the variables xi... |

62 |
Oriented matroids
- Folkman, Lawrence
- 1978
(Show Context)
Citation Context ...ne-Dress theorem; it was announced by Dress, proved in the unpublished dissertation of Bohne and reproved in [10]. By the standard bijection between oriented matroids and pseudoline arrangements (see =-=[9]-=-), rhombus tilings are therefore in bijection with certain pseudo-line arrangements; we use this perspective occasionally (Lemmas 3.1 and 3.2) 1 . Finally, tilings are in bijection with certain commut... |

49 |
The Laurent phenomenon
- Fomin, Zelevinsky
(Show Context)
Citation Context ...polynomial in the initial variables xi,j,k, i.e. a polynomial in the variables xi,j,k and x-1i,j,k. Sergey Fomin and Andrei Zelevinsky, using techniques from the theory of cluster algebras, proved in =-=[4]-=- that the recurrence again generates Laurent polynomials for a large variety of other initial sets (i.e., sets of points (i, j, k) for which we designate fi,j,k = xi,j,k). In [10], David Speyer showed... |

38 |
Determinants and alternating sign matrices
- Robbins, Rumsey
- 1986
(Show Context)
Citation Context ...of graphs but also as enumerating compatible pairs of alternating-sign matrices: in each monomial, the exponents with which the variables xi,j,k appear correspond to the entries of the matrices. (See =-=[9]-=- for details.) In particular, if we set xi,j,k = 1 whenever k = -1, then the terms of the resulting polynomials (in the variables xi,j,0) correspond precisely to single alternating-sign matrices. As P... |

31 | Perfect matchings and the octahedron recurrence
- Speyer
- 2007
(Show Context)
Citation Context ...ry of cluster algebras to give combinatorial formulas for the above mentioned Laurent polynomials. In the case of the octahedron recurrence with n = 3, this problem was solved by the second author in =-=[13]-=-. In addition to the appearance of elegant algebraic varieties, the multidimensional octahedron recurrence has a connection to the representation theory of GLm. To see this, consider the tropical vers... |

30 | Applications of graphical condensation for enumerating matchings and tilings
- Kuo
(Show Context)
Citation Context ...s determined by the shape of I. For suitable initial sets, the corresponding graphs include the Aztec diamond graphs (whose perfect matchings correspond to the domino tilings of Aztec diamonds -- see =-=[7]-=-) and, more generally, the pine-cone graphs of [1]. Consequently, we have a bijection between the set of perfect matchings of a graph (determined by I) and a particular subset of the groves on the ini... |

29 | Higher Bruhat orders and cyclic hyperplane arrangements. Topology 32
- Ziegler
- 1993
(Show Context)
Citation Context ...ation classes of reduced words in the symmetric group SP ai ; see [3]. These commutation classes, in the case a1 = a2 = ... = an = 1, correspond to the elements of the higher Bruhat order B(n,2); see =-=[16]-=-. The main results of this section are Corollary 3.11 (downward flips will eventually lead to Tmin) with its corollary Proposition 3.13 (the graph of flips is connected); Proposition 3.14 (describing ... |

28 |
Combinatorics of Coxeter groups, Graduate Texts
- Björner, Brenti
- 2005
(Show Context)
Citation Context ...mma, we will sometimes identify tilings with their projection under π. Let I ∈ Π, let j < k < ℓ be three numbers between 1 and n, and let us assume that the cube c = { } I + xej + yek + zeℓ ∣ x,y,z ∈ =-=[0,1]-=- (7) is contained in C. The three facets of c containing I + ek are then called the bottom faces of c and those containing I + ej + eℓ the top faces. If a tiling T contains the top faces of c then the... |

26 | A positive proof of the Littlewood-Richardson rule using the octahedron recurrence, Elect
- Knutson, Woodward
(Show Context)
Citation Context ...xI+ej+ek + xI+eℓ+ei , (4) xI+ei+ek + xI+ej+eℓ = max( xI+ei+ej and introduce the following inequalities: s t u v xI + xI+ei−ek ≥ xI+ei−ej + xI+ej−ek . (5) A hive is a solution to (5) in Z ∆(2,m) , see =-=[8]-=-. In other words, it is a triangular array of integers subject to the above inequalities. The recurrence (4) turns out to propagate these inequalities. This fact was then used by Knutson, Tao, and Woo... |

24 | The many faces of alternating-sign matrices
- Propp
(Show Context)
Citation Context ...as connections with the Hirota equation in physics, with Dodgson's condensation method of evaluating determinants, with alternating-sign matrices, and with domino tilings of Aztec diamonds (see [11], =-=[8]-=-, [9], [3], the electronic journal of combinatorics 11 (2004), #R73 1srespectively). It turns out that every fi,j,k is a Laurent polynomial in the initial variables xi,j,k, i.e. a polynomial in the va... |

23 | Extension spaces of oriented matroids - Sturmfels, Ziegler - 1993 |

22 |
Rhombic tilings of polygons and classes of reduced words in Coxeter groups
- Elnitsky
- 1997
(Show Context)
Citation Context ...line arrangements; we use this perspective occasionally (Lemmas 3.1 and 3.2) 1 . Finally, tilings are in bijection with certain commutation classes of reduced words in the symmetric group SP ai ; see =-=[3]-=-. These commutation classes, in the case a1 = a2 = ... = an = 1, correspond to the elements of the higher Bruhat order B(n,2); see [16]. The main results of this section are Corollary 3.11 (downward f... |

19 | A survey of Hirota’s difference equations
- Zabrodin
- 1997
(Show Context)
Citation Context ...hich has connections with the Hirota equation in physics, with Dodgson's condensation method of evaluating determinants, with alternating-sign matrices, and with domino tilings of Aztec diamonds (see =-=[11]-=-, [8], [9], [3], the electronic journal of combinatorics 11 (2004), #R73 1srespectively). It turns out that every fi,j,k is a Laurent polynomial in the initial variables xi,j,k, i.e. a polynomial in t... |

16 | Connected components in the intersection of two open opposite Schubert cells - Shapiro, Shapiro, et al. |

15 |
Teilungen der Ebene durch Geraden oder topologische Geraden
- Ringel
- 1955
(Show Context)
Citation Context ... Proposition 3.13 has been established many times. The first proof in the context of rhombus tilings may be due to Kenyon [7, Theorem 5]; Ringel gave a proof in the context of pseudoline arrangements =-=[11]-=-. The more detailed Corollary 3.11 has been proved in the context of higher Bruhat orders as the statement that B(n,2) has a unique minimal element. Proposition 3.14 can be deduced from a result on or... |

13 | The octahedron recurrence and gln crystals
- Henriques, Kamnitzer
(Show Context)
Citation Context ..., in order to identify the Littlewood-Richardson coefficients with the number of hives subject to certain boundary conditions. Their computation was later refined by the first author and by Kamnitzer =-=[6]-=- in order to describe the associator in the category of gl m-crystals. The case n = 4 of the recurrence is related to the fact that the associator in this category satisfies the pentagon axiom. Our mo... |

9 |
D.: A new approach to solving three combinatorial enumeration problems on planar graphs
- Colbourn, Provan, et al.
- 1995
(Show Context)
Citation Context ...tive proof earlier in part because it elucidates better how acyclicity follows directly from the other properties of a grove. Readers familiar with delta-wye reduction of graphs, as described e.g. in =-=[2]-=-, may notice a resemblance between the correspondence described in Figure 10 and delta-wye moves. Specifically: given I, I0, and (i, j, k) as in the proof of Theorem 1, suppose ^G is the subgraph of G... |

9 | Tiling a Polygon with Parallelograms Algorithmica 9 - Kenyon - 1993 |

6 |
The many faces of alternating-sign matrices. In Discrete models: combinatorics, computation, and geometry
- Propp
- 2001
(Show Context)
Citation Context ...has connections with the Hirota equation in physics, with Dodgson’s condensation method of evaluating determinants, with alternating-sign matrices, and with domino tilings of Aztec diamonds (see [9], =-=[6]-=-, [7], [2], 1respectively). It turns out that every fi,j,k is a Laurent polynomial in the initial xi,j,k, i.e. a polynomial in the variables xi,j,k, x −1 i,j,k . Sergey Fomin and Andrei Zelevinsky, u... |

5 |
Mathematical entertainments: The strange and surprising saga of the somos sequences
- Gale
- 1991
(Show Context)
Citation Context ...+ j + k = n), and it is then satisfied by sn = 3bn 2/4c, by an easy induction. The second and third, when we set sn = 1 for 1 <= n <= 6 (resp. 1 <= n <= 7), are the Somos-6 and Somos-7 sequences (see =-=[6]-=-), which are also really specializations of the cube recurrence. These sequences' definitions are simple enough that it would not be surprising to see them naturally crop up elsewhere, and the combina... |

3 |
Perfect matchings and the octahedron recurrence, preprint
- Speyer
- 2003
(Show Context)
Citation Context ...r algebras, proved in [4] that the recurrence again generates Laurent polynomials for a large variety of other initial sets (i.e., sets of points (i, j, k) for which we designate fi,j,k = xi,j,k). In =-=[10]-=-, David Speyer showed further that all such polynomials could be interpreted as enumerating perfect matchings of suitable bipartite planar graphs, generalizing the main result of [3]. In the present p... |

2 |
A survey of Hirota’s Difference equations, Theoret
- Zabrodin
- 1997
(Show Context)
Citation Context ...hich has connections with the Hirota equation in physics, with Dodgson’s condensation method of evaluating determinants, with alternating-sign matrices, and with domino tilings of Aztec diamonds (see =-=[11]-=-, [8], [9], [3], the electronic journal of combinatorics 11 (2004), #R73 1srespectively). It turns out that every fi,j,k is a Laurent polynomial in the initial variables xi,j,k, i.e. a polynomial in t... |

2 |
The Cube Recurrence Elec
- Carroll, Speyer
(Show Context)
Citation Context ...hese relations as the multidimensional cube recurrence. The use of the term “recurrence” will become clear in Section 2. In the case where n = 3, this was studied in unpublished work of Propp, and in =-=[2]-=-. Y (A) denote the set of solutions of these equations in (C×) Qn i=1 (ai+1) = (C×) Π(A) . Call an element (i1,...,in) ∈ Π even or odd depending on the parity of i1 + · · · + in, and let (C×) 2 act on... |

2 |
Zonotopal tilings and the Bohne-Dress theorem Contemp
- Richter-Gebert, Ziegler
- 1994
(Show Context)
Citation Context ...complex of C which projects homeomorphically onto T . Lemma 2.1 is essentially a special case of the equivalence between “weak zonotopal tilings” and “strong zonotopal tilings” proven in section 3 of =-=[10]-=-. In view of the above lemma, we will sometimes identify tilings with their projection under π. Let I ∈ Π, let j < k < ℓ be three numbers between 1 and n, and let us assume that the cube c = { } I + x... |

1 |
Perfect matchings for Gale-Robinson sequences” (joint work with
- Bousquet-Mélou
- 2002
(Show Context)
Citation Context ...al sets, the corresponding graphs include the Aztec diamond graphs (whose perfect matchings correspond to the domino tilings of Aztec diamonds -- see [7]) and, more generally, the pine-cone graphs of =-=[1]-=-. Consequently, we have a bijection between the set of perfect matchings of a graph (determined by I) and a particular subset of the groves on the initial set I. Addressing this correspondence here wo... |

1 |
Perfect Matchings and the Octahedron Recurrence," preprint, available at http://www.arxiv.org/abs/math.CO/0402452
- Speyer
(Show Context)
Citation Context ...r algebras, proved in [4] that the recurrence again generates Laurent polynomials for a large variety of other initial sets (i.e., sets of points (i, j, k) for which we designate fi,j,k = xi,j,k). In =-=[10]-=-, David Speyer showed further that all such polynomials could be interpreted as enumerating perfect matchings of suitable bipartite planar graphs, generalizing the main result of [3]. In the present p... |

1 |
The Many Faces of Alternating-Sign
- Propp
- 2001
(Show Context)
Citation Context ...as connections with the Hirota equation in physics, with Dodgson’s condensation method of evaluating determinants, with alternating-sign matrices, and with domino tilings of Aztec diamonds (see [11], =-=[8]-=-, [9], [3], the electronic journal of combinatorics 11 (2004), #R73 1srespectively). It turns out that every fi,j,k is a Laurent polynomial in the initial variables xi,j,k, i.e. a polynomial in the va... |