# Bruhat decomposition for general linear group of degree four over a finite field

This article describes the details of the Bruhat decomposition for the general linear group of degree four over a field $K$. Let $G =GL(4,K)$ 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:S4. 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 four-dimensional space $K^4$, 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 $(1,4)(2,3)$.

## Finite field case

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

$G/B \to W$