Grassmannians and Cluster Algebras

By Joshua S. Scott. 2003

396 citations recorded by the source corpus.

This paper follows the program of study initiated by S. Fomin and A. Zelevinsky, and demonstrates that the homogeneous coordinate ring of the Grassmannian G(k, n) is a cluster algebra of geometric type. Those Grassmannians that are of finite cluster type are identified and their cluster variables are interpreted geometrically in terms of configurations of points in C P2. 2000 Mathematics Subject Classification 22E46, 05Exx.

Subfields: Algebra and Number Theory, Food Science, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f5f73f21a9725bda10d502fe129e3a16132bd5b8

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. B. Keller
2. I. Reiten
3. Bernard Leclerc
4. Ralf Schiffler
5. Aslak Bakke Buan
6. Osamu Iyama
7. C. Geiss
8. J. Schröer

In Food Science

1. S. Cecotti
2. Roger Casals
3. C. Vafa
4. Philippe Caldero
5. C. Vergu
6. Michael Shapiro
7. Bernard Leclerc
8. J. Golden

In Geometry and Topology

1. B. Keller
2. I. Reiten
3. Bernard Leclerc
4. Ralf Schiffler
5. Osamu Iyama
6. Aslak Bakke Buan
7. C. Geiss
8. J. Schröer

Citations in the corpus

• Total positivity, Grassmannians, and networks (2006, methodology): “Scott [Sco1, Sco2 ] used our alternating strand diagrams to study Leclerc-Zelevinsky’s [LZ] quasi-commuting families of quantum minors and cluster algebra on the Grassmannian.”
• Dimer models and cluster categories of Grassmannians (2013, methodology): “A class of Postnikov diagrams was used by Scott [27] to show that the homogeneous coordinate ring of Gr(k, n) is a cluster algebra, in which each such diagram corresponds to a seed whose (extended) cluster consists of minors (that is, Plücker coordinates).”
• The twist for positroid varieties (2016, background): “Scott’s result was first presented in her dissertation in 2001 [32], and was placed on the arXiv as a separate paper in 2003 [33] (publication date 2006).”
• 2-frieze patterns and the cluster structure of the space of polygons (2010, neutral): “However, we do not know if there is a nice way to characterize them using Scott’s approach (in terms of Postnikov arrangement or root correspondence).”
• Positroid varieties and cluster algebras (2019, background): “The non-trivial part of (2) is that unlike in the case of the top-dimensional positroid variety [Sco06], not every Plücker coordinate is the image of a cluster variable.”
• Cohomology of cluster varieties, I: Locally acyclic case (2016, background): “ Gr(r,d) of r-planes in Cd is obtained by removing the nhyperplanes {∆ i,i+1,...,i+r−1 = 0} from Gr(r,d) [KLS], where ∆I denotes a Plu¨cker coordinate. The open positroid variety is a cluster variety [Sco]. The open positroid variety ˚Π(2 ,d) ⊂ Gr(2,d) turns out to be a cluster variety A(B˜) of real”
• Weak separation and plabic graphs (2011, methodology): “xpected that the coordinate ring of the Grassmannian, in its Plu¨cker embedding, would be a cluster algebra, and that the Plu¨cker coordinates would be cluster variables. This was verified by Scott in [11]. Moreover, Scott showed that, given any reduced alternating strand diagram, the face labels of ”
• R-Twisting and 4d/2d Correspondences (2010, methodology): “Moreover, comparing the corresponding singularities, we have the identifications in table 2 for the Grassmannian cluster– algebras of finite–type [45, 49, 65] which naturally follows from the CY 3-fold description.”
• Triangulated Categories: Cluster algebras, quiver representations and triangulated categories (2008, positive): “The homogeneous coordinate ring of the Grassmannian Gr(k, n) itself also has a cluster algebra structure [ 120 ] and so have partial flag varieties, double Bruhat cells, Schubert varieties ..., cf. [63] [17].”
• Tensor diagrams and cluster algebras (2012, methodology): “Cluster structures in these rings [21, 48] play a prominent role in applications of cluster theory arising in connection with integrable systems, algebraic Lie theory, Poisson geometry, Teichmüller theory, total positivity, and beyond; see, e.”
• Positivity and canonical bases in rank 2 cluster algebras of finite and affine types (2003, neutral): “• Grassmannians and projective configurations [16].”
• A Formula for Plücker Coordinates Associated with a Planar Network (2008, background): “Scott [10], and for the exploration of tropical analogues by D.”
• Tilting theory and cluster combinatorics (2004, background): “These algebras were defined so that the cluster structure (when quantised) should encode multiplicative properties of the dual canonical basis of the quantised enveloping algebra of a semisimple Lie algebra over C, and that it should model the (classical and quantised) coordinate rings of varieties ”
• Positive geometries and canonical forms (2017, methodology): “For example, the cluster structure of G(k, n) is established in [12].”