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

{{group family-specific information|
information type = Bruhat decomposition|
group family = general linear group of degree two|
connective = of}}

This article describes the details of the [[Bruhat decomposition]] for the [[general linear group of degree two]] over a [[field]] <math>K</math>. Let <math>G =GL(2,K)</math> and <math>B</math> denote the [[Borel subgroup of general linear group|Borel subgroup]] of upper-triangular matrices. We have that:

<math>G = \bigcup_{w \in W} BwB</math>

where <math>W</math> 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'' <math>G \to W</math> whose fibers are the [[double coset]]s of <math>B</math>, and whose restriction to the subgroup <math>W</math> of <math>G</math> is the identity map. The map is well defined because every double coset of <math>B</math> intersects <math>W</math> at a unique point.

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

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

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

==Interpretation in terms of flags==

The mapping:

<math>G/B \to W</math>

can be interpreted as follows: an element of <math>G/B</math> is a complete flag of subspaces for the two-dimensional space <math>K^2</math>, and the mapping to <math>W</math> describes its ''relative position'' with respect to the standard flag (the one stabilized by <math>B</math>). If the flag is equal to the standard flag, then the map sends it to the identity element of <math>W</math>, otherwise it is sent to the non-identity element of <math>W</math>.

==Finite field case==

In the finite field case, for a [[finite field]] with <math>q</math> elements, the fibers for the Bruhat map:

<math>G/B \to W</math>

have sizes 1 and <math>q</math> respectively. More explicitly:

{| class="sortable" border="1"
! Element of <math>W</math> !! Expression as Bruhat word !! Matrix !! Size of fiber in <math>G/B</math> !! Degree of polynomial (= Bruhat length) !! Size of fiber in <math>G</math> (equals <math>|B|</matH> times size of fiber in <math>G/B</math>)!! Explanation 
|-
| <math>()</math> || empty word || <math>\begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix}</math> || 1 || 0 || <math>q(q-1)^2</math> || only the flag stabilized by <math>B</math>.
|-
| <math>(1,2)</math> || <math>s_1</math> || <math>\begin{pmatrix} 0 & 1 \\ 1 & 0 \\\end{pmatrix}</math> || <math>q</math> || 1 || <math>q^2(q-1)^2</math> || Number of one-dimensional subspaces other than the one used in the flag stabilized by <math>B</math>, becomes <math>(q + 1) - 1 = q</math>.
|-
! Total !! --!! -- !! <math>q + 1</math> = size of <math>G/B</math> !! -- !! <math>q(q + 1)(q - 1)^2</math> = order of <matH>GL(2,q)</math> !! --
|}

==Related==

* [[Bruhat decomposition for general linear group of degree three over a field]]
* [[Bruhat decomposition for general linear group of degree two over a discrete valuation ring]]