The pentagram map and Y-patterns

By Max Glick. 2010

76 citations recorded by the source corpus.

The pentagram map, introduced by R. Schwartz, is defined by the following construction: given a polygon as input, draw all of its "shortest" diagonals, and output the smaller polygon which they cut out. We employ the machinery of cluster algebras to obtain explicit formulas for the iterates of the pentagram map.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0c9bbaac6e594adfbf2edb6642570de3fbb7dd70

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. Sergey Fomin
2. Ralf Schiffler
3. Andrei Zelevinsky
4. Lauren K. Williams
5. Michael Shapiro
6. Bernard Leclerc
7. B. Keller
8. Serge Tabachnikov

In Computational Theory and Mathematics

1. Serge Tabachnikov
2. Valentin Ovsienko
3. Max Glick
4. Michael Shapiro
5. A. Vainshtein
6. Michael I. Gekhtman
7. Sergey Fomin
8. Andrei Zelevinsky

In Geometry and Topology

1. Sergey Fomin
2. Ralf Schiffler
3. Andrei Zelevinsky
4. Lauren K. Williams
5. Serge Tabachnikov
6. Michael Shapiro
7. Max Glick
8. Valentin Ovsienko

Citations in the corpus

• Liouville–Arnold integrability of the pentagram map on closed polygons (2011, background): “This relation was noticed in [15] and developed in [7], where the pentagram map on the space of twisted n-gons is interpreted as a sequence of cluster algebra mutations, and an explicit formula for the iterations of T is calculated1.”
• Introduction to Cluster Algebras (2018, background): “e relationship to many areas of mathematics. Here is a partial list of related topics: combinatorics [39], hyperbolic geometry [11, 12, 40], Lie theory [19], Poisson geometry [24], integrable systems [9, 26], representations of associative algebras [6, 7, 5, 46, 45, 47], mathematical physics [10, 1]”
• Introduction to Cluster Algebras. Chapters 1-3 (2016, methodology): “ctive plane P2) defined by the following construction: given a polygonA asinput, drawall of its“shortest” diagonals, and output the “smaller” polygon A′ which they cut out. See Figure 3.7. As shown in [30], the pentagram map is related to Y-seed mutation. To explain the connection, one needs to descr”
• Higher pentagram maps, weighted directed networks, and cluster dynamics (2011, methodology): “A related higher dimensional generalization of the pentagram map was considered by M. Glick [12].”
• INTEGRABLE CLUSTER DYNAMICS OF DIRECTED NETWORKS AND PENTAGRAM MAPS (2014, unknown): “Note that T (1) 3 is the pentagram map on Pn/R∗ considered by Glick [16].”