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 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 Matrix Size of fiber in G/B Size of fiber in G
() \begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix} 1 q(q-1)^2
(1,2) \begin{pmatrix} 0 & 1 \\ 1 & 0 \\\end{pmatrix} q q^2(q-1)^2

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