Given a semisimple linear algebraic $k$-group $G$, one has a spherical building $Delta_G$, and one can interpret the geometric realisation $Delta_G(mathbb R)$ of $Delta_G$ in terms of cocharacters of $G$. The aim of this paper is to extend this construction to the case when $G$ is an arbitrary connected linear algebraic group; we call the resulting object $Delta_G(mathbb R)$ the spherical edifice of $G$. We also define an object $V_G(mathbb R)$ which is an analogue of the vector building for a semisimple group; we call $V_G(mathbb R)$ the vector edifice. The notions of a linear map and an isomorphism between edifices are introduced; we construct some linear maps arising from natural group-theoretic operations. We also devise a family of metrics on $V_G(mathbb R)$ and show they are all bi-Lipschitz equivalent to each other; with this extra structure, $V_G(mathbb R)$ becomes a complete metric space. Finally, we present some motivation in terms of geometric invariant theory and variations on the Tits Centre Conjecture.
{"title":"Edifices: building-like spaces associated to linear algebraic groups","authors":"Michael Bate, Benjamin Martin, Gerhard Röhrle","doi":"10.2140/iig.2023.20.79","DOIUrl":"https://doi.org/10.2140/iig.2023.20.79","url":null,"abstract":"Given a semisimple linear algebraic $k$-group $G$, one has a spherical building $Delta_G$, and one can interpret the geometric realisation $Delta_G(mathbb R)$ of $Delta_G$ in terms of cocharacters of $G$. The aim of this paper is to extend this construction to the case when $G$ is an arbitrary connected linear algebraic group; we call the resulting object $Delta_G(mathbb R)$ the spherical edifice of $G$. We also define an object $V_G(mathbb R)$ which is an analogue of the vector building for a semisimple group; we call $V_G(mathbb R)$ the vector edifice. The notions of a linear map and an isomorphism between edifices are introduced; we construct some linear maps arising from natural group-theoretic operations. We also devise a family of metrics on $V_G(mathbb R)$ and show they are all bi-Lipschitz equivalent to each other; with this extra structure, $V_G(mathbb R)$ becomes a complete metric space. Finally, we present some motivation in terms of geometric invariant theory and variations on the Tits Centre Conjecture.","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":"135690279","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 text is dedicated to Jacques Tits's ideas on geometry over F1, the field with one element. In a first part, we explain how thin Tits geometries surface as rational point sets over the Krasner hyperfield, which links these ideas to combinatorial flag varieties in the sense of Borovik, Gelfand and White and F1-geometry in the sense of Connes and Consani. A completely novel feature is our approach to algebraic groups over F1 in terms of an alteration of the very concept of a group. In the second part, we study an incidence-geometrical counterpart of (epimorphisms to) thin Tits geometries; we introduce and classify all F1-structures on 3-dimensional projective spaces over finite fields. This extends recent work of Thas and Thas on epimorphisms of projective planes (and other rank 2 buildings) to thin planes.
{"title":"Towards the horizons of Tits’s vision : on band schemes, crowds and 𝔽1-structures","authors":"Oliver Lorscheid, Koen Thas","doi":"10.2140/iig.2023.20.353","DOIUrl":"https://doi.org/10.2140/iig.2023.20.353","url":null,"abstract":"This text is dedicated to Jacques Tits's ideas on geometry over F1, the field with one element. In a first part, we explain how thin Tits geometries surface as rational point sets over the Krasner hyperfield, which links these ideas to combinatorial flag varieties in the sense of Borovik, Gelfand and White and F1-geometry in the sense of Connes and Consani. A completely novel feature is our approach to algebraic groups over F1 in terms of an alteration of the very concept of a group. In the second part, we study an incidence-geometrical counterpart of (epimorphisms to) thin Tits geometries; we introduce and classify all F1-structures on 3-dimensional projective spaces over finite fields. This extends recent work of Thas and Thas on epimorphisms of projective planes (and other rank 2 buildings) to thin planes.","PeriodicalId":36589,"journal":{"name":"Innovations in Incidence Geometry","volume":"360 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":"135690287","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 finite graph with an assignment of non-negative integers to vertices gives chip-firing games. Chip-firing games determine languages (sets of words) called the record sets of legal games. Bjorner, Lovasz and Shor found several properties that are satisfied by record sets. In this paper, we will find two more properties of record sets. Under the assumption that the record set is finite and the game fires only two vertices, these properties characterize the record sets of graphs.
{"title":"Finite record sets of chip-firing games","authors":"Kentaro Akasaka, Suguru Ishibashi, Masahiko Yoshinaga","doi":"10.2140/iig.2023.20.55","DOIUrl":"https://doi.org/10.2140/iig.2023.20.55","url":null,"abstract":"A finite graph with an assignment of non-negative integers to vertices gives chip-firing games. Chip-firing games determine languages (sets of words) called the record sets of legal games. Bjorner, Lovasz and Shor found several properties that are satisfied by record sets. In this paper, we will find two more properties of record sets. Under the assumption that the record set is finite and the game fires only two vertices, these properties characterize the record sets of graphs.","PeriodicalId":36589,"journal":{"name":"Innovations in Incidence Geometry","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135980657","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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