Grassmannian-parameterized solutions to direct-sum polygon and simplex equations

Abstract

We consider polygon and simplex equations, of which the simplest nontrivial examples are pentagon (5-gon) and Yang--Baxter (2-simplex), respectively. We examine the general structure of (2n+1)-gon and 2n-simplex equations in direct sums of vector spaces. Then we provide a construction for their solutions, parameterized by elements of the Grassmannian Gr(n+1,2n+1).

arXiv comment26 pages, 5 tikz figures. v4: Introduction re-written, Discussion added in the end

References in corpus (7)

• Permutation-type solutions to the Yang-Baxter and other n-simplex equations
• Simplex and Polygon Equations
• Matrix KP: tropical limit and Yang-Baxter maps
• Cohomologies of $n$-simplex relations
• Nonconstant hexagon relations and their cohomology
• Pentagon Relations in Direct Sums and Grassmann Algebras
• Polynomial-valued constant hexagon cohomology

Cited by in corpus (6)

• On non-abelian quadrirational Yang-Baxter maps
• Non-Abelian hierarchies of compatible maps, associated integrable difference systems and Yang-Baxter maps
• Matrix factorizations and pentagon maps
• Tetrahedron maps, Yang-Baxter maps, and partial linearisations
• Heptagon relation in a direct sum
• Odd-gon relations and their cohomology