Coxeter embeddings are injective

Abstract

We show that certain embeddings of Coxeter groups within other Coxeter groups are injective using the notion of Coxeter partitions. Moreover, we study Lusztig's partitions, which are generalizations of Lusztig's admissible maps and Crisp's foldings. We show that they classify the simplest type of Coxeter partitions, whose embeddings of Coxeter groups send each generator to a product of commuting generators. Consequently, these embeddings are also injective, and we prove that they preserve Coxeter numbers. These results were previously known, due to work of Mühlherr and Dyer.

arXiv commentv2: Thanks to some helpful leads, we found that this paper contains no new results (albeit providing shorter proofs in some cases). We leave this paper on the arXiv to make the results more accessible, but do not intend to publish it. Discussion of prior work and the corresponding references can be found in the final section. v1: 8 pages. Comments welcomed!