Bernhard M¨uhlherr, Holger P. Petersson, and Richard M. Weiss
- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691166902
- eISBN:
- 9781400874019
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691166902.003.0011
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics
This chapter deals with the case that the building at infinity of the Bruhat-Tits building Ξ is a Moufang quadrangle of type E⁶, E₇, and E₈. It begins with a hypothesis that takes into account a ...
More
This chapter deals with the case that the building at infinity of the Bruhat-Tits building Ξ is a Moufang quadrangle of type E⁶, E₇, and E₈. It begins with a hypothesis that takes into account a quadratic space of type Eℓ for ℓ = 6, 7 or 8, K which is complete with respect to a discrete valuation, the two residues of Ξ, and the two root group sequences of a Moufang polygon. It then considers the case that Ξ is an unramified quadrangle if the proposition δΨ = 2 holds. It also explains two other propositions: Ξ is a semi-ramified quadrangle if δΛ = 1 and δΨ = 2 holds, and a ramified quadrangle if δΛ = δΨ = 1 holds.Less
This chapter deals with the case that the building at infinity of the Bruhat-Tits building Ξ is a Moufang quadrangle of type E⁶, E₇, and E₈. It begins with a hypothesis that takes into account a quadratic space of type Eℓ for ℓ = 6, 7 or 8, K which is complete with respect to a discrete valuation, the two residues of Ξ, and the two root group sequences of a Moufang polygon. It then considers the case that Ξ is an unramified quadrangle if the proposition δΨ = 2 holds. It also explains two other propositions: Ξ is a semi-ramified quadrangle if δΛ = 1 and δΨ = 2 holds, and a ramified quadrangle if δΛ = δΨ = 1 holds.
Bernhard M¨uhlherr, Holger P. Petersson, and Richard M. Weiss
- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691166902
- eISBN:
- 9781400874019
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691166902.003.0014
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics
This chapter summarizes the different cases about Moufang quadrangles of type E⁶, E₇ and E₈. The first case is that the building at infinity of the Bruhat-Tits building Ξ is an unramified ...
More
This chapter summarizes the different cases about Moufang quadrangles of type E⁶, E₇ and E₈. The first case is that the building at infinity of the Bruhat-Tits building Ξ is an unramified quadrangle; the second, a semi-ramified quadrangle; and the third, a ramified quadrangle. The chapter considers a theorem that takes into account two root group sequences, both of which are either indifferent or the various dimensions, types, etc., are as indicated in exactly one of twenty-three cases. It also presents a number of propositions relating to a quaternion division algebra and a quadratic space of type Eℓ for ℓ = 6, 7 or 8. Finally, it emphasizes the fact that the quadrangles of type F₄ could have been overlooked in the classification of Moufang polygons.Less
This chapter summarizes the different cases about Moufang quadrangles of type E⁶, E₇ and E₈. The first case is that the building at infinity of the Bruhat-Tits building Ξ is an unramified quadrangle; the second, a semi-ramified quadrangle; and the third, a ramified quadrangle. The chapter considers a theorem that takes into account two root group sequences, both of which are either indifferent or the various dimensions, types, etc., are as indicated in exactly one of twenty-three cases. It also presents a number of propositions relating to a quaternion division algebra and a quadratic space of type Eℓ for ℓ = 6, 7 or 8. Finally, it emphasizes the fact that the quadrangles of type F₄ could have been overlooked in the classification of Moufang polygons.
