nLab
BN-pair

Contents

Contents

Definition

A pair B,NB,N of subgroups of a group GG is a BN-pair if the following holds:

  • the union BNB\cup N generates GG,

  • the intersection BNB\cap N is a normal subgroup of NN,

  • the quotient group WN/(BN)W \coloneqq N/(B\cap N) is generated by a set SS of involutions,

  • for each sS,wWs\in S,w\in W we have BsBBwBBswBBwBB s B \cdot B w B\subseteq B s w B\cup B w B

  • for sSs\in S, BsBBsBBB s B\cdot B s B \neq B

Note that the set SS is uniquely determined by these axioms, and that (W,S)(W,S) is a Coxeter system.

References

Last revised on October 25, 2020 at 09:54:13. See the history of this page for a list of all contributions to it.