{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:18Z","timestamp":1753893858721,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Boyd (1974) proposed a class of infinite ball packings that are generated by\u00a0inversions. Later, Maxwell (1983) interpreted Boyd's construction in terms\u00a0of root systems in Lorentz spaces. In particular, he showed that the\u00a0space-like weight vectors correspond to a ball packing if and only if the\u00a0associated Coxeter graph is of \"level 2\"'. In Maxwell's work, the simple\u00a0roots form a basis of the representations space of the Coxeter group. In\u00a0several recent studies, the more general based root system is considered,\u00a0where the simple roots are only required to be positively independent. In\u00a0this paper, we propose a geometric version of \"level'' for root systems to\u00a0replace Maxwell's graph theoretical \"level''. Then we show that Maxwell's\u00a0results naturally extend to the more general root systems with positively\u00a0independent simple roots. In particular, the space-like extreme rays of the\u00a0Tits cone correspond to a ball packing if and only if the root system is of\u00a0level $2$. We also present a partial classification of level-$2$ root\u00a0systems, namely the Coxeter $d$-polytopes of level-$2$ with $d+2$ facets.<\/jats:p>","DOI":"10.37236\/4989","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T20:41:57Z","timestamp":1578688917000},"source":"Crossref","is-referenced-by-count":1,"title":["Even More Infinite Ball Packings from Lorentzian Root Systems"],"prefix":"10.37236","volume":"23","author":[{"given":"Hao","family":"Chen","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2016,8,5]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v23i3p16\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v23i3p16\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T05:18:00Z","timestamp":1579238280000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v23i3p16"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,8,5]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2016,7,8]]}},"URL":"https:\/\/doi.org\/10.37236\/4989","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2016,8,5]]},"article-number":"P3.16"}}