Strings in abstract root systems

Abstract

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