# Difference between revisions of "Bruhat decomposition for general linear group of degree two over a field"

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 $G$ 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 $(q + 1) - 1 = q$.