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

TL;DR

This paper explores solutions to polygon and simplex equations using Grassmannian parameterization, providing a unified approach to their structure in direct sum vector spaces.

Contribution

It introduces a novel construction for solutions to polygon and simplex equations parameterized by Grassmannian elements, expanding understanding of their algebraic structure.

Findings

Solutions parameterized by Grassmannian elements are explicitly constructed.

The structure of polygon and simplex equations in direct sums is clarified.

A unified framework for these equations is established.

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).