Strings in abstract root systems

TL;DR

This paper introduces a generalized concept of strings within abstract root systems, extending classical notions by considering subsets of simple roots and their linear combinations with elements of the root system.

Contribution

It defines the $\Phi$-string of an element in an abstract root system, broadening the classical understanding of root strings to more general subsets of roots.

Findings

Provides a new framework for analyzing root systems.

Generalizes classical root string concepts.

Lays groundwork for further algebraic investigations.

Abstract

Let $\Phi$ be a subset of the simple roots of a (possibly non-reduced) abstract root system $\Sigma$, and let $\lambda \in \Sigma$. We define the $\Phi$-string of $\lambda$ as the set of elements in $\Sigma \cup \{0\}$ of the form $\lambda + \sum_{\alpha \in \Phi} n_\alpha \alpha$, where $n_\alpha$ is an integer for each $\alpha \in \Phi$. This notion can be regarded as some sort of generalization of the classical notion of $\alpha$-string, where $\alpha \in \Sigma$.