{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:04:33Z","timestamp":1760241873912,"version":"build-2065373602"},"reference-count":12,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2018,9,28]],"date-time":"2018-09-28T00:00:00Z","timestamp":1538092800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We consider certain     E n    -type root lattices embedded within the standard Lorentzian lattice      Z  n + 1     ( 3 \u2264 n \u2264 8 )      and study their discrete geometry from the point of view of del Pezzo surface geometry. The lattice      Z  n + 1        decomposes as a disjoint union of affine hyperplanes which satisfy a certain periodicity. We introduce the notions of line vectors, rational conic vectors, and rational cubics vectors and their relations to E-polytopes. We also discuss the relation between these special vectors and the combinatorics of the Gosset polytopes of type      ( n \u2212 4 )  21    .<\/jats:p>","DOI":"10.3390\/sym10100443","type":"journal-article","created":{"date-parts":[[2018,9,28]],"date-time":"2018-09-28T13:37:58Z","timestamp":1538141878000},"page":"443","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Lorentzian Lattices and E-Polytopes"],"prefix":"10.3390","volume":"10","author":[{"given":"Adrian","family":"Clingher","sequence":"first","affiliation":[{"name":"Department of Mathematics and Computer Science, University of Missouri-St. Louis, St. Louis, MO 63121, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jae-Hyouk","family":"Lee","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Ewha Womans University 52, Ewhayeodae-gil, Seodaemun-gu, Seoul 03760, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,9,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1011","DOI":"10.1155\/S1073792898000609","article-title":"Coxeter groups, Lorentzian lattices, and K3 surfaces","volume":"19","author":"Borcherds","year":"1998","journal-title":"Int. Math. Res. Notices"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1215\/S0012-7094-00-10424-3","article-title":"Reflection groups of Lorentzian lattices","volume":"104","author":"Borcherds","year":"2000","journal-title":"Duke Math. J."},{"key":"ref_3","unstructured":"Dolgachev, I.V. (2018, July 24). Topics in Classical Algebraic Geometry. Available online: http:\/\/www.math.lsa.umich.edu\/\\char126\\relaxidolga\/lecturenotes.html."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Demazure, M., Pinkham, H., and Teissier, B. (1980). Surfaces de Del Pezzo I, II, III, IV, V. S\u00e9minaire sur les Singularit\u00e9s des Surfaces, Springer. Lecture Notes in Mathematics 777.","DOI":"10.1007\/BFb0085872"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"123","DOI":"10.4153\/CJM-2011-063-6","article-title":"Gosset Polytopes in Picard Groups of del Pezzo Surfaces","volume":"64","author":"Lee","year":"2012","journal-title":"Canad. J. Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"4939","DOI":"10.1090\/S0002-9947-2014-06098-4","article-title":"Configurations of lines in del Pezzo surfaces with Gosset Polytopes","volume":"366","author":"Lee","year":"2014","journal-title":"Trans. Amer. Math. Soc."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Lee, J.H., and Shin, Y.J. (2016). E-Polytopes in Picard Groups of Smooth Rational Surfaces. Symmetry, 8.","DOI":"10.3390\/sym8040027"},{"key":"ref_8","unstructured":"Barvinok, A. (2018, July 24). Combinatorics, Geometry and Complexity of Integer Points. Available online: http:\/\/www. math.lsa.umich.edu\/barvinok\/latticenotes669.pdf."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Conway, J.H., and Sloane, N.J.A. (1999). Sphere Packings, Lattices and Groups, Springer. [3rd ed.]. Grundlehren der mathematischen Wissenschaften.","DOI":"10.1007\/978-1-4757-6568-7"},{"key":"ref_10","unstructured":"Conway, J.H. (2008). The Symmetries of Things, A.K. Peters Ltd."},{"key":"ref_11","unstructured":"Coxeter, H.S.M. (1973). Regular Polytopes, Dover Publication Inc.. [3rd ed.]."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1263","DOI":"10.1216\/RMJ-2016-46-4-1263","article-title":"Contaction of del Pezzo surfaces to P2 or P1 \u00d7 P1","volume":"46","author":"Lee","year":"2016","journal-title":"Rocky Mountain J. Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/10\/10\/443\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:23:02Z","timestamp":1760196182000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/10\/10\/443"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,9,28]]},"references-count":12,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2018,10]]}},"alternative-id":["sym10100443"],"URL":"https:\/\/doi.org\/10.3390\/sym10100443","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2018,9,28]]}}}