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

This article gives specific information, namely, Bruhat decomposition, about a family of groups, namely: general linear group of degree two.
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This article describes the details of the Bruhat decomposition for the general linear group of degree two over a field $K$. Let $G=GL(2,K)$ 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 is the symmetric group of degree two (which is isomorphic to cyclic group:Z2). 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.

In this case, $W$ is a set of size two, so there are only two double cosets, one being the subgroup $B$ itself (all elements in this map to the identity element of $W$ and the other being the elements of $G$ outside $B$ (these map to the non-identity element of $W$).

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 two-dimensional space $K^{2}$, 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 the non-identity element of $W$.

Finite field case

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

$G/B\to W$

have sizes 1 and $q$ respectively. More explicitly:

Element of $W$	Expression as Bruhat word	Matrix	Size of fiber in $G/B$	Degree of polynomial (= Bruhat length)	Size of fiber in $G$ (equals $|B|$ times size of fiber in $G/B$)	Explanation
$()$	empty word	${\begin{pmatrix}1&0\\0&1\\\end{pmatrix}}$	1	0	$q(q-1)^{2}$	only the flag stabilized by $B$.
$(1,2)$	$s_{1}$	${\begin{pmatrix}0&1\\1&0\\\end{pmatrix}}$	$q$	1	$q^{2}(q-1)^{2}$	Number of one-dimensional subspaces other than the one used in the flag stabilized by $B$, becomes $(q+1)-1=q$.
Total	--	--	$q+1$ = size of $G/B$	--	$q(q+1)(q-1)^{2}$ = order of $GL(2,q)$	--

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• Bruhat decomposition for general linear group of degree two over a discrete valuation ring