Bruhat decomposition for general linear group of degree three over a field

This article describes the details of the Bruhat decomposition for the general linear group of degree three over a field $$K$$. Let $$G =GL(3,K)$$ and $$B$$ denote the Borel subgroup of upper-triangular matrices. We have that:

$$G = \bigcup_{w \in W} BwB$$

where $$W$$ is the Weyl group, which in this case can be identified with symmetric group:S3. In other words, there is a set map $$G \to W$$ whose fibers are the double cosets of $$B$$, and whose restriction to the subgroup $$W$$ of $$G$$ is the identity map. The map is well defined because every double coset of $$B$$ intersects $$W$$ at a unique point.

Note that the set map is not a homomorphism of groups.

Another way of putting this is that there is a set map from the left coset space $$G/B$$ to $$W$$ that sends a left coset containing an element of $$W$$ to that element of $$W$$, and that is invariant under the left action of $$B$$ by multiplication.

Interpretation in terms of flags
The mapping:

$$G/B \to W$$

can be interpreted as follows: an element of $$G/B$$ is a complete flag of subspaces for the three-dimensional space $$K^3$$, and the mapping to $$W$$ describes its relative position with respect to the standard flag (the one stabilized by $$B$$). If the flag is equal to the standard flag, then the map sends it to the identity element of $$W$$, otherwise it is sent to one of the non-identity element of $$W$$. The generic flag gets sent to the anti-diagonal permutation, corresponding to $$(1,3)$$.

Finite field case
In the finite field case, for a finite field with $$q$$ elements, the fibers for the Bruhat map:

$$G/B \to W$$

can be computed explicitly.