$Y$ -meshes and generalized pentagram maps

Authors :: Pavlo Pylyavskyy
Max Glick
School of Mathematics (UMN-MATH)
University of Minnesota [Twin Cities] (UMN)
University of Minnesota System-University of Minnesota System
James Haglund
Jiang Zeng
Source :: Discrete Mathematics and Theoretical Computer Science, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015), 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015), Jul 2015, Daejeon, South Korea. pp.169-180
Publication Year :: 2015
Publisher :: Centre pour la Communication Scientifique Directe (CCSD), 2015.
Abstract: 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. Our framework incorporates many preexisting generalized pentagram maps due to M. Gekhtman, M. Shapiro, S. Tabachnikov, and A. Vainshtein and also B. Khesin and F. Soloviev. In several of these cases a reduction to cluster dynamics was not previously known.<br />48 pages, 22 figures, to appear in Proceedings of the London Mathematical Society