Root Graded Groups

• Applicant: Professor Dr. Bernhard Mühlherr
• Subject Area: Mathematics
• Term: from 2019 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 428606039

• Final Report Year: 2025

Final Report Abstract

The notion of a root graded group appears for the first time in a paper of Shi from 1993. In that paper he provides a complete classification for the irreducible root gradings of simply laced type of rank at least 3. For the type Cn (n ≥ 3) Zhang obtained some partial results in his PhD-thesis from 2014. In the project, the classification of irreducible crystallographic root gradings of rank at least 3 was almost completed. Furthermore, classification results were obtained in the rank 2 case under suitable additional hypothesises. The results obtained in the project rely on methods that were developed by Tits in the context of his theory of Moufang buildings. The outcome of the project yields a geometric perspective on root gradings of groups. In particular, they provide new methods for investigating Chevalley groups over commutative rings of stable range 1. Furthermore, several results about pseudo-quadratic spaces over stable rings with involution were obtained.

Publications

• Dagger-sharp Tits octagons. J. Korean Math. Soc. 58 (2021), 173–205
  B. Mühlherr & R.M. Weiss
  
• Orthogonal Tits Quadrangles. New Zealand Journal of Mathematics, 52, 427-452.
  Mühlherr, Bernhard & Weiss, Richard
  
• The Exceptional Tits Quadrangles Revisited. Transformation Groups, 29(4), 1683-1698.
  Mühlherr, Bernhard & Weiss, Richard M.
  
• Veldkamp quadrangles and polar spaces. Indagationes Mathematicae, 33(3), 636-663.
  Mühlherr, Bernhard & Weiss, Richard M.
  
• Les quadrangles de Tits classiques. Bulletin de la Société mathématique de France.
  MÜHLHERR, Bernhard & WEISS, Richard M.
  
• Root Graded Groups. PhD-thesis
  Torben Wiedemann
  
• Stable pseudo-quadratic modules. Journal of Combinatorial Algebra, 10(1), 195-233.
  Mühlherr, Bernhard & Weiss, Richard M.
  
• Tits pentagons and the root system GH2. J. Korean Math. Soc. 61 (2024), no. 6, 1095–1126
  B. Mühlherr & R. Weiss
  
• Root graded groups of type H3 and H4. Journal of Algebra, 682, 481–544.
  Berg, Lennart & Wiedemann, Torben