Normality and smoothness of simple linear group compactifications
Identifieur interne : 000B16 ( Istex/Corpus ); précédent : 000B15; suivant : 000B17Normality and smoothness of simple linear group compactifications
Auteurs : Jacopo Gandini ; Alessandro RuzziSource :
- Mathematische Zeitschrift [ 0025-5874 ] ; 2013-10-01.
Abstract
Abstract: Given a semisimple algebraic group $$G$$ , we characterize the normality and the smoothness of its simple linear compactifications, namely those equivariant $$G\times G$$ -compactifications possessing a unique closed orbit which arise in a projective space of the shape $$\mathbb{P }(\mathrm{End}(V))$$ , where $$V$$ is a finite dimensional rational $$G$$ -module. Both the characterizations are purely combinatorial and are expressed in terms of the highest weights of $$V$$ . In particular, we show that $${\mathrm{Sp}}(2r)$$ (with $$r \geqslant 1$$ ) is the unique non-adjoint simple group which admits a simple smooth compactification.
Url:
DOI: 10.1007/s00209-012-1136-3
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CorrespondingAffiliationID="Aff1"><AuthorName DisplayOrder="Western"><GivenName>Jacopo</GivenName><FamilyName>Gandini</FamilyName></AuthorName><Contact><Email>gandini@mat.uniroma1.it</Email></Contact></Author><Author AffiliationIDS="Aff2"><AuthorName DisplayOrder="Western"><GivenName>Alessandro</GivenName><FamilyName>Ruzzi</FamilyName></AuthorName><Contact><Email>Alessandro.Ruzzi@math.univ-bpclermont.fr</Email></Contact></Author><Affiliation ID="Aff1"><OrgDivision>Dipartimento Di Matematica “Guido Castelnuovo”</OrgDivision><OrgName>“Sapienza” Università Di Roma</OrgName><OrgAddress><Street>Piazzale Aldo Moro 5</Street><Postcode>00185 </Postcode><City>Roma</City><Country Code="IT">Italy</Country></OrgAddress></Affiliation><Affiliation ID="Aff2"><OrgDivision>Laboratoire De Mathématiques</OrgDivision><OrgName>Université Blaise Pascal, UMR 6620</OrgName><OrgAddress><Street>CNRS Campus Des Cézeaux</Street><Postcode>63171 </Postcode><City>Aubière Cedex</City><Country Code="FR">France</Country></OrgAddress></Affiliation></AuthorGroup><Abstract ID="Abs1" Language="En" OutputMedium="All"><Heading>Abstract</Heading><Para>Given a semisimple algebraic group <InlineEquation ID="IEq1"><InlineMediaObject><ImageObject Color="BlackWhite" FileRef="209_2012_1136_Article_IEq1.gif" Format="GIF" Rendition="HTML" Type="Linedraw"></ImageObject></InlineMediaObject><EquationSource Format="TEX">$$G$$</EquationSource></InlineEquation>, we characterize the normality and the smoothness of its simple linear compactifications, namely those equivariant <InlineEquation ID="IEq2"><InlineMediaObject><ImageObject Color="BlackWhite" FileRef="209_2012_1136_Article_IEq2.gif" Format="GIF" Rendition="HTML" Type="Linedraw"></ImageObject></InlineMediaObject><EquationSource Format="TEX">$$G\times G$$</EquationSource></InlineEquation>-compactifications possessing a unique closed orbit which arise in a projective space of the shape <InlineEquation ID="IEq3"><InlineMediaObject><ImageObject Color="BlackWhite" FileRef="209_2012_1136_Article_IEq3.gif" Format="GIF" Rendition="HTML" Type="Linedraw"></ImageObject></InlineMediaObject><EquationSource Format="TEX">$$\mathbb{P }(\mathrm{End}(V))$$</EquationSource></InlineEquation>, where <InlineEquation ID="IEq4"><InlineMediaObject><ImageObject Color="BlackWhite" FileRef="209_2012_1136_Article_IEq4.gif" Format="GIF" Rendition="HTML" Type="Linedraw"></ImageObject></InlineMediaObject><EquationSource Format="TEX">$$V$$</EquationSource></InlineEquation> is a finite dimensional rational <InlineEquation ID="IEq5"><InlineMediaObject><ImageObject Color="BlackWhite" FileRef="209_2012_1136_Article_IEq5.gif" Format="GIF" Rendition="HTML" Type="Linedraw"></ImageObject></InlineMediaObject><EquationSource Format="TEX">$$G$$</EquationSource></InlineEquation>-module. Both the characterizations are purely combinatorial and are expressed in terms of the highest weights of <InlineEquation ID="IEq6"><InlineMediaObject><ImageObject Color="BlackWhite" FileRef="209_2012_1136_Article_IEq6.gif" Format="GIF" Rendition="HTML" Type="Linedraw"></ImageObject></InlineMediaObject><EquationSource Format="TEX">$$V$$</EquationSource></InlineEquation>. In particular, we show that <InlineEquation ID="IEq7"><InlineMediaObject><ImageObject Color="BlackWhite" FileRef="209_2012_1136_Article_IEq7.gif" Format="GIF" Rendition="HTML" Type="Linedraw"></ImageObject></InlineMediaObject><EquationSource Format="TEX">$${\mathrm{Sp}}(2r)$$</EquationSource></InlineEquation> (with <InlineEquation ID="IEq8"><InlineMediaObject><ImageObject Color="BlackWhite" FileRef="209_2012_1136_Article_IEq8.gif" Format="GIF" Rendition="HTML" Type="Linedraw"></ImageObject></InlineMediaObject><EquationSource Format="TEX">$$r \geqslant 1$$</EquationSource></InlineEquation>) is the unique non-adjoint simple group which admits a simple smooth compactification.</Para></Abstract></ArticleHeader><NoBody></NoBody></Article></Issue></Volume></Journal></Publisher></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>Normality and smoothness of simple linear group compactifications</title></titleInfo><titleInfo type="alternative" contentType="CDATA" lang="en"><title>Normality and smoothness of simple linear group compactifications</title></titleInfo><name type="personal" displayLabel="corresp"><namePart type="given">Jacopo</namePart><namePart type="family">Gandini</namePart><affiliation>Dipartimento Di Matematica “Guido Castelnuovo”, “Sapienza” Università Di Roma, Piazzale Aldo Moro 5, 00185, Roma, Italy</affiliation><affiliation>E-mail: gandini@mat.uniroma1.it</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">Alessandro</namePart><namePart type="family">Ruzzi</namePart><affiliation>Laboratoire De Mathématiques, Université Blaise Pascal, UMR 6620, CNRS Campus Des Cézeaux, 63171, Aubière Cedex, France</affiliation><affiliation>E-mail: Alessandro.Ruzzi@math.univ-bpclermont.fr</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre type="research-article" displayLabel="OriginalPaper" authority="ISTEX" authorityURI="https://content-type.data.istex.fr" valueURI="https://content-type.data.istex.fr/ark:/67375/XTP-1JC4F85T-7">research-article</genre><originInfo><publisher>Springer Berlin Heidelberg</publisher><place><placeTerm type="text">Berlin/Heidelberg</placeTerm></place><dateCreated encoding="w3cdtf">2012-03-28</dateCreated><dateIssued encoding="w3cdtf">2013-10-01</dateIssued><dateIssued encoding="w3cdtf">2012</dateIssued><copyrightDate encoding="w3cdtf">2012</copyrightDate></originInfo><language><languageTerm type="code" authority="rfc3066">en</languageTerm><languageTerm type="code" authority="iso639-2b">eng</languageTerm></language><abstract lang="en">Abstract: Given a semisimple algebraic group $$G$$ , we characterize the normality and the smoothness of its simple linear compactifications, namely those equivariant $$G\times G$$ -compactifications possessing a unique closed orbit which arise in a projective space of the shape $$\mathbb{P }(\mathrm{End}(V))$$ , where $$V$$ is a finite dimensional rational $$G$$ -module. 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