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Division algebras and transitivity of group actions on buildings
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Matthew C. B. Zaremsky
Published/Copyright:
April 3, 2015
Abstract
Let D be a division algebra with center F and degree d > 2. Let K|F be any splitting field.We analyze the action of D× and SL1(D) on the spherical and affine buildings that may be associated to GLd(K) and SLd(K), and in particular show it is never strongly transitive. In the affine case we find examples where the action is nonetheless Weyl transitive. This extends results of Abramenko and Brown concerning the d = 2 case, where strong transitivity is in fact possible. Our approach produces some explicit constructions, and we find that for d > 2 the failure of the action to be strongly transitive is quite dramatic.
Received: 2013-5-7
Accepted: 2013-6-11
Published Online: 2015-4-3
Published in Print: 2015-4-1
© 2015 by Walter de Gruyter Berlin/Boston
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Articles in the same Issue
- Frontmatter
- Division algebras and transitivity of group actions on buildings
- Projections of del Pezzo surfaces and Calabi–Yau threefolds
- Almost soliton duality
- Remarks on Kähler–Ricci solitons
- Real solutions to systems of polynomial equations and parameter continuation
- Division pairs: a new approach to Moufang sets
- Very special divisors on 4-gonal real algebraic curves
- On the intersection of a Hermitian surface with an elliptic quadric
- Geometric properties of semitube domains
- Threefolds in ℙ6 of degree 12
Articles in the same Issue
- Frontmatter
- Division algebras and transitivity of group actions on buildings
- Projections of del Pezzo surfaces and Calabi–Yau threefolds
- Almost soliton duality
- Remarks on Kähler–Ricci solitons
- Real solutions to systems of polynomial equations and parameter continuation
- Division pairs: a new approach to Moufang sets
- Very special divisors on 4-gonal real algebraic curves
- On the intersection of a Hermitian surface with an elliptic quadric
- Geometric properties of semitube domains
- Threefolds in ℙ6 of degree 12
