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

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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: | 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: | ||

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{| class="sortable" border="1" | {| class="sortable" border="1" | ||

− | ! Element of <math>W</math> !! Matrix !! Size of fiber in <math>G/B</math> !! Size of fiber in <math>G</math> | + | ! 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>()</math> || <math>\begin{pmatrix} 1 | + | | <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> !! -- | |

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## Latest revision as of 02:50, 31 March 2012

This article gives specific information, namely, Bruhat decomposition, about a family of groups, namely: general linear group of degree two.

View Bruhat decomposition of group families | View other specific information about general linear group of degree two

This article describes the details of the Bruhat decomposition for the general linear group of degree two 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 is the symmetric group of degree two (which is isomorphic to cyclic group:Z2). 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.

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

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 two-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 the non-identity element of .

## Finite field case

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

have sizes 1 and respectively. More explicitly:

Element of | Expression as Bruhat word | Matrix | Size of fiber in | Degree of polynomial (= Bruhat length) | Size of fiber in (equals times size of fiber in ) | Explanation |
---|---|---|---|---|---|---|

empty word | 1 | 0 | only the flag stabilized by . | |||

1 | Number of one-dimensional subspaces other than the one used in the flag stabilized by , becomes . | |||||

Total | -- | -- | = size of | -- | = order of | -- |