The affine Weyl group symmetry of Desargues maps and of the non-commutative Hirota-Miwa system

TL;DR

This paper explores the geometric and algebraic symmetries of Desargues maps and the non-commutative Hirota-Miwa system, revealing affine Weyl group invariance and its implications for integrable systems.

Contribution

It provides a geometric characterization of Desargues maps invariant under affine Weyl group actions, linking them to the non-commutative Hirota-Miwa system.

Findings

Desargues maps are characterized by collinearity of images of N-simplex vertices.

Affine Weyl group symmetry underpins the invariance of the non-commutative Hirota-Miwa system.

The geometric interpretation offers new insights into the symmetries of integrable systems.

Abstract

We study recently introduced Desargues maps, which provide simple geometric interpretation of the non-commutative Hirota--Miwa system. We characterize them as maps of the A-type root lattice into a projective space such that images of vertices of any basic regular N-simplex are collinear. Such a characterization is manifestly invariant with respect to the corresponding affine Weyl group action, which leads to related symmetries of the Hirota--Miwa system.