{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:44:13Z","timestamp":1787323453521,"version":"build-2736575974"},"reference-count":16,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Discrete Math."],"published-print":{"date-parts":[[2009,1]]},"abstract":"<jats:p>For a rank two root system and a pair of nonnegative integers, using only elementary combinatorics we construct two posets. The constructions are uniform across the root systems $A_{1}\\oplus A_{1}$, $A_{2}$, $C_{2}$, and $G_{2}$. Examples appear in Figures Four and Five. We then form the distributive lattices of order ideals of these posets. Corollary CharacterCorollary gives elegant quotient-of-products expressions for the rank generating functions of these lattices (thereby providing answers to a 1979 question of Stanley). Also, Theorem CharacterProposition describes how these lattices provide a new combinatorial setting for the Weyl characters of representations of rank two semisimple Lie algebras. Most of these lattices are new; the rest of them (or related structures) have arisen in the work of Stanley, Kashiwara, Nakashima, Littelmann, and Molev. In a future paper, one author shows that the posets constructed here form a Dynkin diagram-indexed answer to a combinatorially posed classification question. In a companion paper, some of these lattices are used to explicitly construct some representations of rank two semisimple Lie algebras. This implies that these lattices are strongly Sperner.<\/jats:p>","DOI":"10.1137\/070689887","type":"journal-article","created":{"date-parts":[[2009,2,4]],"date-time":"2009-02-04T18:13:34Z","timestamp":1233771214000},"page":"527-559","source":"Crossref","is-referenced-by-count":2,"title":["Distributive Lattices Defined for Representations of Rank Two Semisimple Lie Algebras"],"prefix":"10.1137","volume":"23","author":[{"suffix":"II","given":"L. Wyatt","family":"Alverson","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Robert G.","family":"Donnelly","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Scott J.","family":"Lewis","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Marti","family":"McClard","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Robert","family":"Pervine","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Robert A.","family":"Proctor","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"N. J.","family":"Wildberger","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2009,2,4]]},"reference":[{"key":"R1","unstructured":"L. W. Alverson II,\n                      Distributive Lattices and Representations of the Rank Two Simple Lie Algebras\n                      , Master's thesis, Murray State University, Murray, KY, 2003."},{"key":"R2","doi-asserted-by":"crossref","unstructured":"L. W. Alverson II, R. G. Donnelly, S. J. Lewis, and R. Pervine,\n                      Constructions of representations of rank two semisimple Lie algebras with distributive lattices\n                      , Electron. J. Combin., 13 (2006), R109 (44 pp).","DOI":"10.37236\/1135"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1023\/A:1025096704917"},{"key":"R4","unstructured":"R. G. Donnelly,\n                      Dynkin diagram classification results satisfying certain structural properties\n                      , in preparation."},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1016\/S0012-365X(02)00523-X"},{"key":"R6","unstructured":"R. G. Donnelly and N. J. Wildberger,\n                      Distributive lattice models for certain families of irreducible semisimple Lie algebra representations\n                      , in preparation."},{"key":"R7","doi-asserted-by":"crossref","unstructured":"J. E. Humphreys,\n                      Introduction to Lie Algebras and Representation Theory\n                      , Springer-Verlag, New York, 1972.","DOI":"10.1007\/978-1-4612-6398-2"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1006\/jabr.1994.1114"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1016\/0021-8693(90)90086-4"},{"key":"R10","unstructured":"M. McClard,\n                      Picturing Representations of Simple Lie Algebras of Rank Two\n                      , Master's thesis, Murray State University, Murray, KY, 2000."},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1016\/S0195-6698(84)80037-2"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(90)90032-R"},{"key":"R13","unstructured":"R. P. Stanley,\n                      Unimodal sequences arising from Lie algebras\n                      , in Young Day Proceedings, T. V. Narayana et al., eds., Marcel Dekker, New York, 1980, pp. 127\u2013136."},{"key":"R14","doi-asserted-by":"crossref","unstructured":"R. P. Stanley,\n                      Enumerative Combinatorics, Vol.\n                      1, Wadsworth and Brooks\/Cole, Monterey, CA, 1986.","DOI":"10.1007\/978-1-4615-9763-6_1"},{"key":"R15","doi-asserted-by":"crossref","unstructured":"R. P. Stanley,\n                      Enumerative Combinatorics, Vol.\n                      2, Cambridge University Press, Cambridge, UK, 1999.","DOI":"10.1017\/CBO9780511609589"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1006\/aima.2001.2050"}],"container-title":["SIAM Journal on Discrete Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/070689887","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:03:13Z","timestamp":1787320993000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/070689887"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,1]]},"references-count":16,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2009,1]]}},"alternative-id":["10.1137\/070689887"],"URL":"https:\/\/doi.org\/10.1137\/070689887","relation":{},"ISSN":["0895-4801","1095-7146"],"issn-type":[{"value":"0895-4801","type":"print"},{"value":"1095-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,1]]}}}