Discrete monodromy, pentagrams, and the method of condensation

By Richard Evan Schwartz. 2007

64 citations recorded by the source corpus.

Abstract.This paper studies the pentagram map, a projectively natural iteration on the space of polygons. Inspired by a method from the theory of ordinary differential equations, the paper constructs roughly n algebraically independent invariants for the map, when it is defined on the space of n-gons. These invariants strongly suggest that the pentagram map is a discrete completely integrable system. The paper also relates the pentagram map to Dodgson’s method of condensation for computing determinants, also known as the octahedral recurrence.

Subfields: Computational Theory and Mathematics, Numerical Analysis, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/5f2182e3aa991be6523084f2b43f945e0038ea8d

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Citations in the corpus

Higher pentagram maps, weighted directed networks, and cluster dynamics (2011, background): “[30] established algebraic-geometric integrability of the pentagram map by constructing its Lax (zero curvature) representation; his approach works both for twisted and closed polygons. It wasshownin [27] that the pentagrammap was intimately relatedto the socalled octahedral recurrence. It was conje”
The pentagram map and Y-patterns (2010, background): “L’application pentagramme de R. Schwartz est définie par laconstruction suivante: on trace les diagonales “les plus courtes” d’un polygone donné en entrée et on retou ne en sortie le plus petit polygone que ces diagonales découpent.”
The Pentagram Map: A Discrete Integrable System (2008, methodology): “In the first coordinate system, which comes from [21], most of the formulas are simpler.”
Introduction to Cluster Algebras (2018, background): “s. Table 1 gives the 8 corresponding conserved quantities for the pentagram map. The ﬁrst 2 are trivial but the remaining 6 are (up to a sign) the original conserved quantities discovered by Schwartz [49] by diﬀerent means. The invariant Poisson bracket from Proposition 6.9 and the conserved quantit”
INTEGRABLE CLUSTER DYNAMICS OF DIRECTED NETWORKS AND PENTAGRAM MAPS (2014, unknown): “ent (i) of Theorem 5.6, along with Theorem 4.12, implies that the generalized higher pentagram map Fk is completely integrable. Integrals of the pentagram map (1.1) were constructed by R. Schwartz in [36]. He observed that the map commutes with the scaling Xi → λXi, Yi → λ −1Y i, and that the conjug”
Liouville–Arnold integrability of the pentagram map on closed polygons (2011, methodology): “d nis odd, which is a usual, generic, situation for an integrable system. If nis even, then d= 1 2 dimC n 1 so that one can talk of \hyper-integrability". Our approach is based on the results of [17] and [13]. We prove that the level sets of the monodromy invariants on the subspace C nˆP n are ”
2-frieze patterns and the cluster structure of the space of polygons (2010, background): “:::;y n 4) restrict to a bijection between R 2n 8 >0 and F 0 n . Theorem 2 provides good coordinate systems for the study of the space C n, dierent from the known coordinate systems described in [26, 20]. We think that the \restricted" set of cluster coordinates Zis an interesting object tha”