The Coxeter relations and KP map for non-commuting symbols

TL;DR

This paper explores the action of the symmetric group on non-commuting variables via the non-Abelian KP system, providing explicit formulas and connections to formal language theory.

Contribution

It introduces a novel group action on non-commuting indeterminates derived from the non-Abelian KP system, with explicit formulas in the periodic case.

Findings

Explicit formulas for the group action in the periodic reduction.

Connection established between the KP map and context-free languages.

Framework linking Coxeter relations with non-commutative integrable systems.

Abstract

We give an action of the symmetric group on non-commuting indeterminates in terms of series in the corresponding Mal'cev-Newmann division ring. The action is constructed from the non-Abelian Hirota-Miwa (discrete KP) system. The corresponding companion map, which gives generators of the action, is discussed in the generic case and the corresponding explicit formulas have been found in the periodic reduction. We discuss also briefly connection of the companion to the KP map with context-free languages.