scholarly journals Weak Separation, Pure Domains and Cluster Distance

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Miriam Farber ◽  
Pavel Galashin
Keyword(s):  
Simple Formula ◽  
Cluster Algebra ◽  
Algebra Structure ◽  
Linear Projection ◽  
Cluster Distance ◽  
International Audience ◽  
Separated Sets ◽  
Weakly Separated Sets ◽  
Weak Separation ◽  
Octahedron Recurrence

International audience Following the proof of the purity conjecture for weakly separated sets, recent years have revealed a variety of wider classes of pure domains in different settings. In this paper we show the purity for domains consisting of sets that are weakly separated from a pair of “generic” sets I and J. Our proof also gives a simple formula for the rank of these domains in terms of I and J. This is a new instance of the purity phenomenon which essentially differs from all previously known pure domains. We apply our result to calculate the cluster distance and to give lower bounds on the mutation distance between cluster variables in the cluster algebra structure on the coordinate ring of the Grassmannian. Using a linear projection that relates weak separation to the octahedron recurrence, we also find the exact mutation distances and cluster distances for a family of cluster variables.

scholarly journals Arrangements of equal minors in the positive Grassmannian

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Miriam Farber ◽  
Alexander Postnikov
Keyword(s):  
Cluster Algebra ◽  
The Other ◽  
Eulerian Number ◽  
Other Hand ◽  
Totally Positive ◽  
Positive Matrices ◽  
International Audience ◽  
Separated Sets ◽  
Weakly Separated Sets ◽  
Totally Positive Matrices

International audience We discuss arrangements of equal minors in totally positive matrices. More precisely, we would like to investigate the structure of possible equalities and inequalities between the minors. We show that arrangements of equals minors of largest value are in bijection with <i>sorted sets</i>, which earlier appeared in the context of <i>alcoved polytopes</i> and Gröbner bases. Maximal arrangements of this form correspond to simplices of the alcoved triangulation of the hypersimplex; and the number of such arrangements equals the <i>Eulerian number</i>. On the other hand, we conjecture and prove in many cases that arrangements of equal minors of smallest value are exactly the <i>weakly separated sets</i>. Weakly separated sets, originally introduced by Leclerc and Zelevinsky, are closely related to the \textitpositive Grassmannian and the associated <i>cluster algebra</i>.

scholarly journals Combinatorics of Positroids

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Suho Oh
Keyword(s):  
Special Class ◽  
Combinatorial Description ◽  
Combinatorial Objects ◽  
International Audience ◽  
Separated Sets ◽  
Weakly Separated Sets

International audience Recently Postnikov gave a combinatorial description of the cells in a totally-nonnegative Grassmannian. These cells correspond to a special class of matroids called positroids. There are many interesting combinatorial objects associated to a positroid. We introduce some recent results, including the generalization and proof of the purity conjecture by Leclerc and Zelevinsky on weakly separated sets.

scholarly journals The purity of set-systems related to Grassmann necklaces

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Vladimir Danilov ◽  
Alexander Karzanov ◽  
Gleb Koshevoy
Keyword(s):  
Boolean Cube ◽  
Symmetric Relation ◽  
Prove Theorem ◽  
Set Systems ◽  
International Audience ◽  
Separated Sets ◽  
Weakly Separated Sets

International audience Studying the problem of quasicommuting quantum minors, Leclerc and Zelevinsky introduced in 1998 the notion of weakly separated sets in $[n]:=\{1,\ldots, n\}$. Moreover, they raised several conjectures on the purity for this symmetric relation, in particular, on the Boolean cube $2^{[n]}$. In 0909.1423[math.CO] we proved these purity conjectures for the Boolean cube $2^{[n]}$, the discrete Grassmanian $\binom{[n]}{r}$, and some other set-systems. Oh, Postnikov, and Speyer in arxiv:1109.4434 proved the purity for weakly separated collections inside a positroid which contain a Grassmann necklace $\mathcal {N}$ defining the positroid. We denote such set-systems as $\mathcal{Int}(\mathcal {N} )$. In this paper we give an alternative (and shorter) proof of the purity of $\mathcal{Int}(\mathcal {N} )$ and present a stronger result. More precisely, we introduce a set-system $\mathcal{Out}(\mathcal {N} )$ complementary to $\mathcal{Int}(\mathcal {N })$, in a sense, and establish its purity. Moreover, we prove (Theorem~3) that these two set-systems are weakly separated from each other. As a consequence of Theorem~3, we obtain the purity of set-systems related to pairs of weakly separated necklaces (Proposition 4 and Corollaries 1 and 2). Finally, we raise a conjecture on the purity of both the interior and exterior of a generalized necklace.

scholarly journals Universal geometric coefficients for the four-punctured sphere (Extended Abstract)

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Emily Barnard ◽  
Emily Meehan ◽  
Shira Polster ◽  
Nathan Reading
Keyword(s):  
Cluster Algebra ◽  
Short List ◽  
Nous Permet ◽  
Rational Part ◽  
International Audience

International audience We construct universal geometric coefficients for the cluster algebra associated to the four-punctured sphere and obtain, as a by-product, the $g$ -vectors of cluster variables. We also construct the rational part of the mutation fan. These constructions rely on a classification of the allowable curves (the curves which can appear in quasi-laminations). The classification allows us to prove the Null Tangle Property for the four-punctured sphere, thus adding this surface to a short list of surfaces for which this property is known. The Null Tangle Property then implies that the shear coordinates of allowable curves are the universal coefficients. We compute these shear coordinates to obtain universal geometric coefficients. Nous construisons des coefficients géométriques universels pour l’algèbre amassée associée à la sphère privée de 4 points, et obtenons ce faisant les $g$-vecteurs des variables d’amas. Nous construisons aussi la partie rationnelle de l’éventail de mutation. Ces constructions reposent sur la classification des courbes admissibles (les courbes qui peuvent apparaître dans les quasi-laminations). Cette classification nous permet de prouver la “Null Tangle Property” pour la sphère privée de 4 points, ajoutant ainsi cette surface à la courte liste de surfaces pour lesquelles cette propriété est connue. La “Null Tangle Property” implique alors que les coordonnées de décalage des courbes admissibles sont les coefficients universels. Nous calculons ces coordonnées de décalage pour obtenir les coefficients géométriques universels.

scholarly journals $Y$ -meshes and generalized pentagram maps

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Max Glick ◽  
Pavlo Pylyavskyy
Keyword(s):  
Projective Geometry ◽  
Cluster Algebra ◽  
Cluster Theory ◽  
Rich Family ◽  
International Audience ◽  
Pentagram Map

International audience We introduce a rich family of generalizations of the pentagram map sharing the property that each generates an infinite configuration of points and lines with four points on each line. These systems all have a description as $Y$ -mutations in a cluster algebra and hence establish new connections between cluster theory and projective geometry. Nous introduisons une famille de généralisations de l’application pentagramme. Chacune produit une configuration infinie de points et de lignes avec quatre points sur chaque ligne. Ces systèmes ont une description des $Y$ -mutations dans une algèbre amassée, un nouveau lien entre la théorie d’algèbres amassées et la géométrie projective.

scholarly journals Minimal factorizations of a cycle: a multivariate generating function

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Philippe Biane ◽  
Matthieu Josuat-Vergès
Keyword(s):  
Symmetric Group ◽  
Generating Function ◽  
Simple Formula ◽  
Hook Length ◽  
Noncrossing Partitions ◽  
Long Cycle ◽  
Number Of Factors ◽  
Multivariate Generalization ◽  
International Audience

International audience It is known that the number of minimal factorizations of the long cycle in the symmetric group into a product of k cycles of given lengths has a very simple formula: it is nk−1 where n is the rank of the underlying symmetric group and k is the number of factors. In particular, this is nn−2 for transposition factorizations. The goal of this work is to prove a multivariate generalization of this result. As a byproduct, we get a multivariate analog of Postnikov's hook length formula for trees, and a refined enumeration of final chains of noncrossing partitions.

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 The cluster and dual canonical bases of Z [x_11, ..., x_33] are equal

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Brendon Rhoades
Keyword(s):  
Quantum Group ◽  
Polynomial Ring ◽  
Geometric Model ◽  
Cluster Algebra ◽  
Canonical Basis ◽  
The Other ◽  
Monomial Basis ◽  
Dual Canonical Basis ◽  
Nous Montrons ◽  
International Audience

International audience The polynomial ring $\mathbb{Z}[x_{11}, . . . , x_{33}]$ has a basis called the dual canonical basis whose quantization facilitates the study of representations of the quantum group $U_q(\mathfrak{sl}3(\mathbb{C}))$. On the other hand, $\mathbb{Z}[x_{11}, . . . , x_{33}]$ inherits a basis from the cluster monomial basis of a geometric model of the type $D_4$ cluster algebra. We prove that these two bases are equal. This extends work of Skandera and proves a conjecture of Fomin and Zelevinsky. This also provides an explicit factorization of the dual canonical basis elements of $\mathbb{Z}[x_{11}, . . . , x_{33}]$ into irreducible polynomials. L'anneau de polynômes $\mathbb{Z}[x_{11}, . . . , x_{33}]$ a une base appelée base duale canonique, et dont une quantification facilite l'étude des représentations du groupe quantique $U_q(\mathfrak{sl}3(\mathbb{C}))$. D'autre part, $\mathbb{Z}[x_{11}, . . . , x_{33}]$ admet une base issue de la base des monômes d'amas de l'algèbre amassée géométrique de type $D_4$. Nous montrons que ces deux bases sont égales. Ceci prolonge les travaux de Skandera et démontre une conjecture de Fomin et Zelevinsky. Ceci fournit également une factorisation explicite en polynômes irréductibles des éléments de la base duale canonique de $\mathbb{Z}[x_{11}, . . . , x_{33}]$ .

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