Bruhat decomposition for general linear group of degree two over a field
where is the Weyl group, which in this case is the symmetric group of degree two (which is isomorphic to cyclic group:Z2). In other words, there is a set map whose fibers are the double cosets of , and whose restriction to the subgroup of is the identity map. The map is well defined because every double coset of intersects at a unique point.
Note that the set map is not a homomorphism of groups.
In this case, is a set of size two, so there are only two double cosets, one being the subgroup itself (all elements in this map to the identity element of and the other being the elements of outside (these map to the non-identity element of ).
Another way of putting this is that there is a set map from the left coset space to that sends a left coset containing an element of to that element of , and that is invariant under the left action of by multiplication.
Interpretation in terms of flags
can be interpreted as follows: an element of is a complete flag of subspaces for the two-dimensional space , and the mapping to describes its relative position with respect to the standard flag (the one stabilized by ). If the flag is equal to the standard flag, then the map sends it to the identity element of , otherwise it is sent to the non-identity element of .
Finite field case
In the finite field case, for a finite field with elements, the fibers for the Bruhat map:
have sizes 1 and respectively. More explicitly:
|Element of||Expression as Bruhat word||Matrix||Size of fiber in||Degree of polynomial (= Bruhat length)||Size of fiber in||Explanation|
|empty word||1||0||only the flag stabilized by .|
|1||Number of one-dimensional subspaces other than the one used in the flag stabilized by , becomes .|
|Total||--||--||--||= order of||--|