Let G be a group. Then G has a BN-pair or a Tits system if the following conditions hold:

  1. 1.

    B and N are subgroupsMathworldPlanetmathPlanetmath of G such that G=<B,N>.

  2. 2.

    BN=TN and N/T=W is a group generated by a set S.

  3. 3.

    sBwBwBBswB for all sS and wW.

  4. 4.

    sBs-1B for all sS.

Where BwB is a double coset with respect to B. It can be proven that S is in fact made up of elements of order 2, and that W is a Coxeter groupMathworldPlanetmath.

Example: Let G=GLn(𝕂) where 𝕂 is some field. Then, if we let B be the subgroup of upper triangular matricesMathworldPlanetmath and N be the subgroup of monomial matrices (i.e. matrices having one nonzero entry in each row and each column, or more precisely the stabilizerMathworldPlanetmath of the lines {[e1],,[en]}). Then, it can be shown that B and N generate G and that T is the subgroup of diagonal matricesMathworldPlanetmath. In turn, it follows that W in this case is isomorphicPlanetmathPlanetmathPlanetmathPlanetmath to the symmetric groupMathworldPlanetmathPlanetmath on n letters, Sn.

For more, consult chapter 5 in the book Buildings, by Kenneth Brown

Title BN-pair
Canonical name BNpair
Date of creation 2013-03-22 15:30:11
Last modified on 2013-03-22 15:30:11
Owner tedgar (10630)
Last modified by tedgar (10630)
Numerical id 11
Author tedgar (10630)
Entry type Definition
Classification msc 20F55
Synonym Tits Systems