Let be a group. Then has a -pair or a Tits system if the following conditions hold:
and are subgroups of such that .
and is a group generated by a set .
for all and .
for all .
Example: Let where is some field. Then, if we let be the subgroup of upper triangular matrices and be the subgroup of monomial matrices (i.e. matrices having one nonzero entry in each row and each column, or more precisely the stabilizer of the lines ). Then, it can be shown that and generate and that is the subgroup of diagonal matrices. In turn, it follows that in this case is isomorphic to the symmetric group on letters, .
For more, consult chapter 5 in the book Buildings, by Kenneth Brown
|Date of creation||2013-03-22 15:30:11|
|Last modified on||2013-03-22 15:30:11|
|Last modified by||tedgar (10630)|