{"title":"Affine buildings: construction by norms and study of isometries","authors":"Anne Parreau","doi":"10.2140/iig.2023.20.471","DOIUrl":"https://doi.org/10.2140/iig.2023.20.471","url":null,"abstract":"","PeriodicalId":36589,"journal":{"name":"Innovations in Incidence Geometry","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135690278","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This is a semi-survey paper, where we start by advertising Tits' synthetic construction from cite{Tits}, of the hyperbolic plane $H^2(Cay)$ over the Cayley numbers $Cay$, and of its automorphism group which is the exceptional simple Lie group $G=F_{4(-20)}$. Let $G=KAN$ be the Iwasawa decomposition. Our contributions are: a) Writing down explicitly the action of $N$ on $H^2(Cay)$ in Tits'model, facing the lack of associativity of $Cay$. b) If $MAN$ denotes the minimal parabolic subgroup of $G$, characterizing $M$ geometrically.
{"title":"The exceptional simple Lie group F4(−20), after J. Tits","authors":"Alain J. Valette","doi":"10.2140/iig.2023.20.599","DOIUrl":"https://doi.org/10.2140/iig.2023.20.599","url":null,"abstract":"This is a semi-survey paper, where we start by advertising Tits' synthetic construction from cite{Tits}, of the hyperbolic plane $H^2(Cay)$ over the Cayley numbers $Cay$, and of its automorphism group which is the exceptional simple Lie group $G=F_{4(-20)}$. Let $G=KAN$ be the Iwasawa decomposition. Our contributions are: a) Writing down explicitly the action of $N$ on $H^2(Cay)$ in Tits'model, facing the lack of associativity of $Cay$. b) If $MAN$ denotes the minimal parabolic subgroup of $G$, characterizing $M$ geometrically.","PeriodicalId":36589,"journal":{"name":"Innovations in Incidence Geometry","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135690268","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Locally projective graphs of symplectic type","authors":"Alexander A. Ivanov","doi":"10.2140/iig.2023.20.303","DOIUrl":"https://doi.org/10.2140/iig.2023.20.303","url":null,"abstract":"","PeriodicalId":36589,"journal":{"name":"Innovations in Incidence Geometry","volume":"154 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135691542","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This is the second part of our study of epimorphisms with source a thick generalized $m$-gon and target a thin generalized $m$-gon. We classify the case $m = 8$ when the polygons are finite (in the first part [15] we handled the cases $m = 3, 4$ and $6$). Then we show that the infinite case is very different, and construct examples which strongly differ from the finite case. A number of general structure theorems are also obtained, and we also take a look at the infinite case for general gonality.
{"title":"Epimorphisms of generalized polygons, B: The octagons","authors":"Joseph A. Thas, Koen Thas","doi":"10.2140/iig.2023.20.579","DOIUrl":"https://doi.org/10.2140/iig.2023.20.579","url":null,"abstract":"This is the second part of our study of epimorphisms with source a thick generalized $m$-gon and target a thin generalized $m$-gon. We classify the case $m = 8$ when the polygons are finite (in the first part [15] we handled the cases $m = 3, 4$ and $6$). Then we show that the infinite case is very different, and construct examples which strongly differ from the finite case. A number of general structure theorems are also obtained, and we also take a look at the infinite case for general gonality.","PeriodicalId":36589,"journal":{"name":"Innovations in Incidence Geometry","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134989790","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Distances between fixed-point sets in metrically complete 2-dimensional Euclidean buildings are realised","authors":"Harris Leung, Jeroen Schillewaert, Anne Thomas","doi":"10.2140/iig.2023.20.341","DOIUrl":"https://doi.org/10.2140/iig.2023.20.341","url":null,"abstract":"","PeriodicalId":36589,"journal":{"name":"Innovations in Incidence Geometry","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135690259","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Triality is a classical notion in geometry that arose in the context of the Lie groups of type $D_4$. Another notion of triality, Wilson triality, appears in the context of reflexible maps. We build a bridge between these two notions, showing how to construct an incidence geometry with a triality from a map that admits a Wilson triality. We also extend a result by Jones and Poulton, showing that for every prime power $q$, the group ${rm L}_2(q^3)$ has maps that admit Wilson trialities but no dualities.
{"title":"Incidence geometries with trialities coming from maps with Wilson trialities","authors":"Dimitri Leemans, Klara Stokes","doi":"10.2140/iig.2023.20.325","DOIUrl":"https://doi.org/10.2140/iig.2023.20.325","url":null,"abstract":"Triality is a classical notion in geometry that arose in the context of the Lie groups of type $D_4$. Another notion of triality, Wilson triality, appears in the context of reflexible maps. We build a bridge between these two notions, showing how to construct an incidence geometry with a triality from a map that admits a Wilson triality. We also extend a result by Jones and Poulton, showing that for every prime power $q$, the group ${rm L}_2(q^3)$ has maps that admit Wilson trialities but no dualities.","PeriodicalId":36589,"journal":{"name":"Innovations in Incidence Geometry","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135690276","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A nonzero element $x$ in a Lie algebra $mathfrak{g}$ with Lie product $[ , ]$ is called extremal if $[x,[x,y]]$ is a multiple of $x$ for all $y$. In this paper we characterize the (finitary) symplectic Lie algebras as simple Lie algebras generated by their extremal elements satisying the condition that any two noncommuting extremal elements $x,y$ generate an $mathfrak{sl}_2$ and any third extremal element $z$ commutes with at least one extremal element in this $mathfrak{sl}_2$.
{"title":"A geometric characterization of the symplectic Lie algebra","authors":"Hans Cuypers, Yael Fleischmann","doi":"10.2140/iig.2023.20.223","DOIUrl":"https://doi.org/10.2140/iig.2023.20.223","url":null,"abstract":"A nonzero element $x$ in a Lie algebra $mathfrak{g}$ with Lie product $[ , ]$ is called extremal if $[x,[x,y]]$ is a multiple of $x$ for all $y$. In this paper we characterize the (finitary) symplectic Lie algebras as simple Lie algebras generated by their extremal elements satisying the condition that any two noncommuting extremal elements $x,y$ generate an $mathfrak{sl}_2$ and any third extremal element $z$ commutes with at least one extremal element in this $mathfrak{sl}_2$.","PeriodicalId":36589,"journal":{"name":"Innovations in Incidence Geometry","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134990405","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper determines the relationship between the geometry of retractions and the combinatorics of folded galleries for arbitrary affine buildings, and so provides a unified framework to study orbits in affine flag varieties. We introduce the notion of labeled folded galleries for any affine building X and use these to describe the preimages of chimney retractions. When X is the building for a group with an affine Tits system, such as the Bruhat-Tits building for a group over a local field, we can then relate labeled folded galleries and shadows to double coset intersections in affine flag varieties. This result generalizes the authors' previous joint work with Naqvi on groups over function fields.
{"title":"Chimney retractions in affine buildings encode orbits in affine flag varieties","authors":"Elizabeth Milićević, Petra Schwer, Anne Thomas","doi":"10.2140/iig.2023.20.395","DOIUrl":"https://doi.org/10.2140/iig.2023.20.395","url":null,"abstract":"This paper determines the relationship between the geometry of retractions and the combinatorics of folded galleries for arbitrary affine buildings, and so provides a unified framework to study orbits in affine flag varieties. We introduce the notion of labeled folded galleries for any affine building X and use these to describe the preimages of chimney retractions. When X is the building for a group with an affine Tits system, such as the Bruhat-Tits building for a group over a local field, we can then relate labeled folded galleries and shadows to double coset intersections in affine flag varieties. This result generalizes the authors' previous joint work with Naqvi on groups over function fields.","PeriodicalId":36589,"journal":{"name":"Innovations in Incidence Geometry","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135690267","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We classify the automorphisms of a Moufang hexagon mapping no chamber to an opposite chamber (such automorphisms are called domestic). This forms part of a larger program to classify domestic automorphisms of Moufang spherical buildings.
{"title":"Automorphisms and opposition in spherical buildings of exceptional type, II: Moufang hexagons","authors":"James Parkinson, Hendrik Van Maldeghem","doi":"10.2140/iig.2023.20.443","DOIUrl":"https://doi.org/10.2140/iig.2023.20.443","url":null,"abstract":"We classify the automorphisms of a Moufang hexagon mapping no chamber to an opposite chamber (such automorphisms are called domestic). This forms part of a larger program to classify domestic automorphisms of Moufang spherical buildings.","PeriodicalId":36589,"journal":{"name":"Innovations in Incidence Geometry","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135690282","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Anton A. Baykalov, Alice Devillers, Cheryl E. Praeger
The sets of primitive, quasiprimitive, and innately transitive permutation groups may each be regarded as the building blocks of finite transitive permutation groups, and are analogues of composition factors for abstract finite groups. This paper extends classifications of finite primitive and quasiprimitive groups of rank at most $3$ to a classification for the finite innately transitive groups. The new examples comprise three infinite families and three sporadic examples. A necessary step in this classification was the determination of certain configurations in finite almost simple $2$-transitive groups called special pairs.
{"title":"Rank three innately transitive permutation groups and related 2-transitive groups","authors":"Anton A. Baykalov, Alice Devillers, Cheryl E. Praeger","doi":"10.2140/iig.2023.20.135","DOIUrl":"https://doi.org/10.2140/iig.2023.20.135","url":null,"abstract":"The sets of primitive, quasiprimitive, and innately transitive permutation groups may each be regarded as the building blocks of finite transitive permutation groups, and are analogues of composition factors for abstract finite groups. This paper extends classifications of finite primitive and quasiprimitive groups of rank at most $3$ to a classification for the finite innately transitive groups. The new examples comprise three infinite families and three sporadic examples. A necessary step in this classification was the determination of certain configurations in finite almost simple $2$-transitive groups called special pairs.","PeriodicalId":36589,"journal":{"name":"Innovations in Incidence Geometry","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135733598","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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