# 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 . Let and denote the Borel subgroup of upper-triangular matrices. We have that:

where is the Weyl group, which in this case can be identified with symmetric group:S3. 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.

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

The mapping:

can be interpreted as follows: an element of is a complete flag of subspaces for the three-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 one of the non-identity element of . The *generic* flag gets sent to the anti-diagonal permutation, corresponding to .

## Finite field case

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

can be computed explicitly.

Element of | Expression in Bruhat terms | Matrix | Size of fiber in (equals to the power of the Bruhat word length) | Degree of polynomial (= word length of Bruhat word) | Size of fiber in | Explanation |
---|---|---|---|---|---|---|

empty word | 1 | 0 | ||||

1 | ||||||

1 | ||||||

2 | ||||||

2 | ||||||

3 | ||||||

Total (6 rows) | -- | -- | equals size of | -- | equals order of |