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\n Let\n \n \n \n G<\/mml:mi>\n G<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n be a simple classical algebraic group over an algebraically closed field\n \n \n \n K<\/mml:mi>\n K<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of characteristic\n \n \n \n \n p<\/mml:mi>\n \n \u2265\n \n <\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n p\\geq 0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n with natural module\n \n \n \n W<\/mml:mi>\n W<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Let\n \n \n \n H<\/mml:mi>\n H<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n be a closed subgroup of\n \n \n \n G<\/mml:mi>\n G<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and let\n \n \n \n V<\/mml:mi>\n V<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n be a nontrivial\n \n \n \n p<\/mml:mi>\n p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -restricted irreducible tensor indecomposable rational\n \n \n \n \n K<\/mml:mi>\n G<\/mml:mi>\n <\/mml:mrow>\n KG<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -module such that the restriction of\n \n \n \n V<\/mml:mi>\n V<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n to\n \n \n \n H<\/mml:mi>\n H<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is irreducible. In this paper we classify the triples\n \n \n \n \n (<\/mml:mo>\n G<\/mml:mi>\n ,<\/mml:mo>\n H<\/mml:mi>\n ,<\/mml:mo>\n V<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n (G,H,V)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of this form, where\n \n \n \n \n V<\/mml:mi>\n \n \u2260\n \n <\/mml:mo>\n W<\/mml:mi>\n ,<\/mml:mo>\n \n W<\/mml:mi>\n \n \n \u2217\n \n <\/mml:mo>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n V \\neq W,W^{*}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n H<\/mml:mi>\n H<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is a disconnected almost simple positive-dimensional closed subgroup of\n \n \n \n G<\/mml:mi>\n G<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n acting irreducibly on\n \n \n \n W<\/mml:mi>\n W<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Moreover, by combining this result with earlier work, we complete the classification of the irreducible triples\n \n \n \n \n (<\/mml:mo>\n G<\/mml:mi>\n ,<\/mml:mo>\n H<\/mml:mi>\n ,<\/mml:mo>\n V<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n (G,H,V)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n where\n \n \n \n G<\/mml:mi>\n G<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is a simple algebraic group over\n \n \n \n K<\/mml:mi>\n K<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , and\n \n \n \n H<\/mml:mi>\n H<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is a maximal closed subgroup of positive dimension.\n <\/p>","DOI":"10.1090\/memo\/1114","type":"journal-article","created":{"date-parts":[[2014,12,16]],"date-time":"2014-12-16T08:58:24Z","timestamp":1418720304000},"source":"Crossref","is-referenced-by-count":7,"title":["Irreducible almost simple subgroups of classical algebraic groups"],"prefix":"10.1090","volume":"236","author":[{"given":"Timothy","family":"Burness","sequence":"first","affiliation":[]},{"given":"Souma\u00efa","family":"Ghandour","sequence":"additional","affiliation":[]},{"given":"Claude","family":"Marion","sequence":"additional","affiliation":[]},{"given":"Donna","family":"Testerman","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2014,12,16]]},"reference":[{"key":"1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1017\/S0027763000017438","article-title":"Involutions in Chevalley groups over fields of even order","volume":"63","author":"Aschbacher, Michael","year":"1976","journal-title":"Nagoya Math. 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