{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:52:28Z","timestamp":1776797548124,"version":"3.51.2"},"reference-count":49,"publisher":"American Mathematical Society (AMS)","issue":"296","license":[{"start":{"date-parts":[[2016,5,8]],"date-time":"2016-05-08T00:00:00Z","timestamp":1462665600000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    In 1979, Kazhdan and Lusztig introduced the notion of \u201ccells\u201d (left, right and two-sided) for a Coxeter group\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper W\">\n                        <mml:semantics>\n                          <mml:mi>W<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">W<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , a concept with numerous applications in Lie theory and around. Here, we address algorithmic aspects of this theory for finite\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper W\">\n                        <mml:semantics>\n                          <mml:mi>W<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">W<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    which are important in applications, e.g., run explicitly through all left cells, determine the values of Lusztig\u2019s\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold a\">\n                        <mml:semantics>\n                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                            <mml:mi mathvariant=\"bold\">a<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbf {a}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -function, identify the characters of left cell representations. The aim is to show how type\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper E 8\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>E<\/mml:mi>\n                            <mml:mn>8<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">E_8<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    (the largest group of exceptional type) can be handled systematically and efficiently, too. This allows us, for the first time, to solve some open questions in this case, including Kottwitz\u2019 conjecture on left cells and involutions.\n                  <\/p>","DOI":"10.1090\/mcom\/2963","type":"journal-article","created":{"date-parts":[[2015,5,8]],"date-time":"2015-05-08T12:52:55Z","timestamp":1431089575000},"page":"3029-3049","source":"Crossref","is-referenced-by-count":4,"title":["On the Kazhdan\u2013Lusztig cells in type \ud835\udc38\u2088"],"prefix":"10.1090","volume":"84","author":[{"given":"Meinolf","family":"Geck","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Abbie","family":"Halls","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2015,5,8]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"160","DOI":"10.1016\/0021-8693(87)90082-2","article-title":"The left cells of the Coxeter group of type \ud835\udc3b\u2084","volume":"107","author":"Alvis, Dean","year":"1987","journal-title":"J. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0021-8693","issn-type":"print"},{"key":"2","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1515\/crll.1982.336.201","article-title":"The representations and generic degrees of the Hecke algebra of type \ud835\udc3b\u2084","volume":"336","author":"Alvis, Dean","year":"1982","journal-title":"J. Reine Angew. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0075-4102","issn-type":"print"},{"key":"3","isbn-type":"print","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1007\/978-0-8176-4697-4_3","article-title":"On domino insertion and Kazhdan-Lusztig cells in type \ud835\udc35_{\ud835\udc5b}","author":"Bonnaf\u00e9, C\u00e9dric","year":"2010","ISBN":"https:\/\/id.crossref.org\/isbn\/9780817646967"},{"key":"4","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1090\/S1088-4165-2014-00456-4","article-title":"Conjugacy classes of involutions and Kazhdan-Lusztig cells","volume":"18","author":"Bonnaf\u00e9, C\u00e9dric","year":"2014","journal-title":"Represent. Theory"},{"key":"5","doi-asserted-by":"publisher","first-page":"32","DOI":"10.1090\/S1088-4165-00-00052-2","article-title":"Verifying Kottwitz\u2019 conjecture by computer","volume":"4","author":"Casselman, Bill","year":"2000","journal-title":"Represent. Theory"},{"issue":"2","key":"6","doi-asserted-by":"publisher","first-page":"805","DOI":"10.1006\/jabr.1999.8376","article-title":"Left cells in the Weyl group of type \ud835\udc38\u2088","volume":"231","author":"Chen, Yu","year":"2000","journal-title":"J. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0021-8693","issn-type":"print"},{"issue":"11","key":"7","doi-asserted-by":"publisher","first-page":"3837","DOI":"10.1080\/00927879208826378","article-title":"Left cells in the Weyl group of type \ud835\udc38\u2087","volume":"26","author":"Chen, Yu","year":"1998","journal-title":"Comm. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0092-7872","issn-type":"print"},{"issue":"3","key":"8","doi-asserted-by":"publisher","first-page":"211","DOI":"10.1007\/BF01190330","article-title":"The state of the art in the computation of Kazhdan-Lusztig polynomials","volume":"7","author":"du Cloux, Fokko","year":"1996","journal-title":"Appl. Algebra Engrg. Comm. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0938-1279","issn-type":"print"},{"issue":"2","key":"9","doi-asserted-by":"publisher","first-page":"731","DOI":"10.1016\/j.jalgebra.2005.10.004","article-title":"Positivity results for the Hecke algebras of noncrystallographic finite Coxeter groups","volume":"303","author":"du Cloux, Fokko","year":"2006","journal-title":"J. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0021-8693","issn-type":"print"},{"issue":"3","key":"10","doi-asserted-by":"publisher","first-page":"1089","DOI":"10.4007\/annals.2014.180.3.6","article-title":"The Hodge theory of Soergel bimodules","volume":"180","author":"Elias, Ben","year":"2014","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"issue":"2","key":"11","first-page":"187","article-title":"On the classification of primitive ideals for complex classical Lie algebras. III","volume":"88","author":"Garfinkle, Devra","year":"1993","journal-title":"Compositio Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-437X","issn-type":"print"},{"issue":"5","key":"12","doi-asserted-by":"publisher","first-page":"608","DOI":"10.1112\/S0024609303002236","article-title":"On the induction of Kazhdan-Lusztig cells","volume":"35","author":"Geck, Meinolf","year":"2003","journal-title":"Bull. London Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6093","issn-type":"print"},{"key":"13","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1090\/S1088-4165-02-00128-0","article-title":"Constructible characters, leading coefficients and left cells for finite Coxeter groups with unequal parameters","volume":"6","author":"Geck, Meinolf","year":"2002","journal-title":"Represent. Theory"},{"key":"14","doi-asserted-by":"publisher","first-page":"481","DOI":"10.1090\/S1088-4165-06-00287-1","article-title":"Relative Kazhdan-Lusztig cells","volume":"10","author":"Geck, Meinolf","year":"2006","journal-title":"Represent. Theory"},{"issue":"3","key":"15","doi-asserted-by":"publisher","first-page":"635","DOI":"10.1017\/S0024611506015930","article-title":"Kazhdan-Lusztig cells and the Murphy basis","volume":"93","author":"Geck, Meinolf","year":"2006","journal-title":"Proc. London Math. Soc. (3)","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6115","issn-type":"print"},{"issue":"3","key":"16","doi-asserted-by":"publisher","first-page":"653","DOI":"10.1017\/S0013091508000394","article-title":"Leading coefficients and cellular bases of Hecke algebras","volume":"52","author":"Geck, Meinolf","year":"2009","journal-title":"Proc. Edinb. Math. Soc. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0013-0915","issn-type":"print"},{"issue":"3","key":"17","doi-asserted-by":"publisher","first-page":"587","DOI":"10.4310\/PAMQ.2011.v7.n3.a5","article-title":"On Iwahori-Hecke algebras with unequal parameters and Lusztig\u2019s isomorphism theorem","volume":"7","author":"Geck, Meinolf","year":"2011","journal-title":"Pure Appl. Math. Q.","ISSN":"https:\/\/id.crossref.org\/issn\/1558-8599","issn-type":"print"},{"issue":"4","key":"18","doi-asserted-by":"publisher","first-page":"905","DOI":"10.4171\/CMH\/273","article-title":"On the Kazhdan-Lusztig order on cells and families","volume":"87","author":"Geck, Meinolf","year":"2012","journal-title":"Comment. Math. Helv.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-2571","issn-type":"print"},{"key":"19","doi-asserted-by":"publisher","first-page":"231","DOI":"10.1112\/S1461157012001064","article-title":"\ud835\uddaf\ud835\uddd2\ud835\udda2\ud835\uddc8\ud835\uddd1: computing with (finite) Coxeter groups and Iwahori-Hecke algebras","volume":"15","author":"Geck, Meinolf","year":"2012","journal-title":"LMS J. Comput. Math."},{"key":"20","doi-asserted-by":"publisher","first-page":"329","DOI":"10.1016\/j.jalgebra.2013.01.019","article-title":"Kazhdan-Lusztig cells and the Frobenius-Schur indicator","volume":"398","author":"Geck, Meinolf","year":"2014","journal-title":"J. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0021-8693","issn-type":"print"},{"key":"21","doi-asserted-by":"crossref","unstructured":"M. Geck, On Kottwitz\u2019 conjecture for twisted involutions, J. Lie Theory 25 (2015), 395\u2013429.","DOI":"10.5802\/jolt.842"},{"key":"22","unstructured":"M. Geck, PyCox \u2013 A Python version of CHEVIE-GAP for (finite) Coxeter groups, version 1r6p180, 2014 (\\url{http:\/\/www.mathematik.uni-stuttgart.de\/ geckmf})."},{"key":"23","unstructured":"M. Geck, A generalised \ud835\udf0f-invariant for the unequal parameter case, preprint, available at arXiv:1405.5736 (see also arXiv:1502.01661)."},{"key":"24","series-title":"Algebra and Applications","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-85729-716-7","volume-title":"Representations of Hecke algebras at roots of unity","volume":"15","author":"Geck, Meinolf","year":"2011","ISBN":"https:\/\/id.crossref.org\/isbn\/9780857297150"},{"issue":"11","key":"25","doi-asserted-by":"publisher","first-page":"3274","DOI":"10.1016\/j.jalgebra.2008.10.024","article-title":"James\u2019 conjecture for Hecke algebras of exceptional type. I","volume":"321","author":"Geck, Meinolf","year":"2009","journal-title":"J. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0021-8693","issn-type":"print"},{"key":"26","series-title":"London Mathematical Society Monographs. New Series","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198502500.001.0001","volume-title":"Characters of finite Coxeter groups and Iwahori-Hecke algebras","volume":"21","author":"Geck, Meinolf","year":"2000","ISBN":"https:\/\/id.crossref.org\/isbn\/0198502508"},{"key":"27","unstructured":"R. B. Howlett, \ud835\udc4a-graphs for the irreducible representations of the Hecke algebra of type \ud835\udc38\u2088, private communication with J. Michel (December 2003)."},{"issue":"8","key":"28","doi-asserted-by":"publisher","first-page":"2055","DOI":"10.1016\/j.jalgebra.2008.12.018","article-title":"Computational construction of irreducible \ud835\udc4a-graphs for types \ud835\udc38\u2086 and \ud835\udc38\u2087","volume":"321","author":"Howlett, Robert B.","year":"2009","journal-title":"J. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0021-8693","issn-type":"print"},{"key":"29","series-title":"Cambridge Studies in Advanced Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511623646","volume-title":"Reflection groups and Coxeter groups","volume":"29","author":"Humphreys, James E.","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/052137510X"},{"issue":"2","key":"30","doi-asserted-by":"publisher","first-page":"165","DOI":"10.1007\/BF01390031","article-title":"Representations of Coxeter groups and Hecke algebras","volume":"53","author":"Kazhdan, David","year":"1979","journal-title":"Invent. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0020-9910","issn-type":"print"},{"key":"31","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1090\/S1088-4165-00-00050-9","article-title":"Involutions in Weyl groups","volume":"4","author":"Kottwitz, Robert E.","year":"2000","journal-title":"Represent. Theory"},{"issue":"2","key":"32","doi-asserted-by":"publisher","first-page":"490","DOI":"10.1016\/0021-8693(81)90188-5","article-title":"On a theorem of Benson and Curtis","volume":"71","author":"Lusztig, George","year":"1981","journal-title":"J. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0021-8693","issn-type":"print"},{"key":"33","series-title":"Annals of Mathematics Studies","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1515\/9781400881772","volume-title":"Characters of reductive groups over a finite field","volume":"107","author":"Lusztig, George","year":"1984","ISBN":"https:\/\/id.crossref.org\/isbn\/0691083509"},{"key":"34","isbn-type":"print","doi-asserted-by":"publisher","first-page":"255","DOI":"10.2969\/aspm\/00610255","article-title":"Cells in affine Weyl groups","author":"Lusztig, George","year":"1985","ISBN":"https:\/\/id.crossref.org\/isbn\/0444877118"},{"issue":"2","key":"35","doi-asserted-by":"publisher","first-page":"536","DOI":"10.1016\/0021-8693(87)90154-2","article-title":"Cells in affine Weyl groups. II","volume":"109","author":"Lusztig, George","year":"1987","journal-title":"J. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0021-8693","issn-type":"print"},{"key":"36","isbn-type":"print","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1090\/pspum\/047.2\/933415","article-title":"Leading coefficients of character values of Hecke algebras","author":"Lusztig, G.","year":"1987","ISBN":"https:\/\/id.crossref.org\/isbn\/0821814788"},{"issue":"1","key":"37","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/S0021-8693(02)00514-8","article-title":"Rationality properties of unipotent representations","volume":"258","author":"Lusztig, G.","year":"2002","journal-title":"J. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0021-8693","issn-type":"print"},{"key":"38","series-title":"CRM Monograph Series","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1090\/crmm\/018","volume-title":"Hecke algebras with unequal parameters","volume":"18","author":"Lusztig, G.","year":"2003","ISBN":"https:\/\/id.crossref.org\/isbn\/0821833561"},{"issue":"3","key":"39","first-page":"323","article-title":"Hecke algebras and involutions in Weyl groups","volume":"7","author":"Lusztig, George","year":"2012","journal-title":"Bull. Inst. Math. Acad. Sin. (N.S.)","ISSN":"https:\/\/id.crossref.org\/issn\/2304-7909","issn-type":"print"},{"key":"40","unstructured":"J. Michel, The development version of the CHEVIE package of GAP3, preprint, available at arXiv:1310.7905."},{"key":"41","unstructured":"H. Naruse, \ud835\udc4a-graphs for the irreducible representations of the Iwahori\u2013Hecke algebras of type \ud835\udc39\u2084 and \ud835\udc38\u2086, private communication with M. Geck (January and July, 1998)."},{"key":"42","series-title":"Lecture Notes in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0074968","volume-title":"The Kazhdan-Lusztig cells in certain affine Weyl groups","volume":"1179","author":"Shi, Jian Yi","year":"1986","ISBN":"https:\/\/id.crossref.org\/isbn\/3540164391"},{"issue":"3","key":"43","doi-asserted-by":"publisher","first-page":"556","DOI":"10.1016\/j.jcta.2005.04.007","article-title":"Left cells containing a fully commutative element","volume":"113","author":"Shi, Jian-Yi","year":"2006","journal-title":"J. Combin. Theory Ser. A","ISSN":"https:\/\/id.crossref.org\/issn\/0097-3165","issn-type":"print"},{"issue":"2","key":"44","doi-asserted-by":"publisher","first-page":"327","DOI":"10.3836\/tjm\/1270132265","article-title":"The left cells and their \ud835\udc4a-graphs of Weyl group of type \ud835\udc39\u2084","volume":"13","author":"Takahashi, Kazuhisa","year":"1990","journal-title":"Tokyo J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0387-3870","issn-type":"print"},{"issue":"13","key":"45","doi-asserted-by":"publisher","first-page":"5031","DOI":"10.1080\/00927879508825517","article-title":"Left cells in Weyl group of type \ud835\udc38\u2086","volume":"23","author":"Tong, Chang Qing","year":"1995","journal-title":"Comm. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0092-7872","issn-type":"print"},{"issue":"3","key":"46","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1007\/BF01420727","article-title":"A generalized \ud835\udf0f-invariant for the primitive spectrum of a semisimple Lie algebra","volume":"242","author":"Vogan, David A., Jr.","year":"1979","journal-title":"Math. Ann.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5831","issn-type":"print"},{"issue":"3","key":"47","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1007\/BF01420525","article-title":"Ordering of the primitive spectrum of a semisimple Lie algebra","volume":"248","author":"Vogan, David A., Jr.","year":"1980","journal-title":"Math. Ann.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5831","issn-type":"print"},{"key":"48","unstructured":"D. A. Vogan, The character table of \ud835\udc38\u2088, Notices of the Amer. Math. Soc. 9 (2007), 1022\u20131034; see also \\url{http:\/\/atlas.math.umd.edu\/AIM_{E}8\/technicaldetails.html}."},{"issue":"6","key":"49","doi-asserted-by":"publisher","first-page":"557","DOI":"10.1112\/blms\/21.6.557","article-title":"An approach to the connectedness of the left cells in affine Weyl groups","volume":"21","author":"Xi, Nan Hua","year":"1989","journal-title":"Bull. London Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6093","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2015-84-296\/S0025-5718-2015-02963-9\/S0025-5718-2015-02963-9.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2015-84-296\/S0025-5718-2015-02963-9\/S0025-5718-2015-02963-9.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:33:37Z","timestamp":1776796417000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2015-84-296\/S0025-5718-2015-02963-9\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,5,8]]},"references-count":49,"journal-issue":{"issue":"296","published-print":{"date-parts":[[2015,11]]}},"alternative-id":["S0025-5718-2015-02963-9"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/2963","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2015,5,8]]}}}