Bruhat decomposition

The Bruhat decomposition is a fundamental method from the theory of algebraic groups . It generalizes the fact, known from the Gaussian elimination method , that every matrix can be decomposed as the product of an upper and lower triangular matrix . The method is named after François Bruhat .

Bruhat decomposition

Let it be a connected reductive algebraic group over an algebraically closed field , a Borel subgroup and the Weyl group of . ${\displaystyle G}$ ${\displaystyle K}$ ${\displaystyle B}$ ${\displaystyle W}$ ${\displaystyle G}$

Then you have a decomposition called a Bruhat decomposition

${\displaystyle G=BWB:=\coprod _{w\in W}BwB}$

of as a disjoint union of double subclasses of parameterized by the elements of the Weyl group . ${\displaystyle G}$ ${\displaystyle B}$ ${\displaystyle W}$

Projective geometry

The double secondary classes correspond to the secondary classes . From the Bruhat decomposition it follows that the Weyl group parameterizes the pairs of elements of the flag variety modulo the action of . ${\displaystyle B\backslash G/B}$ ${\displaystyle G\backslash (G/B\times G/B)}$ ${\displaystyle G}$

In the case of the projective linear group is the plume manifold and from the Bruhat decomposition it follows that there are modulo of the action of exactly pairs of complete plumes . ${\displaystyle PGL(n,K)}$ ${\displaystyle G/B}$ ${\displaystyle PGL(n,K)}$ ${\displaystyle n!}$

example

Let be the projective linear group of the complex matrices. Then the Weyl group consists of two elements, which are represented by the matrices and . Each matrix is ​​therefore a multiple of a matrix that is either of the shape or of the shape, each with triangular matrices . Because then every pair is either in the orbit of or of , where the first case occurs if and only if is with . ${\displaystyle G=PGL(2,\mathbb {C} )}$ ${\displaystyle 2\times 2}$ ${\displaystyle E=\left({\begin{array}{cc}1&0\\0&1\end{array}}\right)}$ ${\displaystyle w_{0}=\left({\begin{array}{cc}0&1\\1&0\end{array}}\right)}$ ${\displaystyle 2\times 2}$ ${\displaystyle LR}$ ${\displaystyle Lw_{0}R}$ ${\displaystyle L,R}$ ${\displaystyle G/B=P^{1}\mathbb {C} =\mathbb {C} ^{2}\setminus \left\{0\right\}/\mathbb {C} ^{*}}$ ${\displaystyle (\left[v\right],\left[w\right])\in P^{1}\mathbb {C} \times P^{1}\mathbb {C} }$ ${\displaystyle PGL(2,\mathbb {C} )}$ ${\displaystyle (\left[e_{1}\right],\left[e_{1}\right])}$ ${\displaystyle (\left[e_{1}\right],\left[e_{2}\right])}$ ${\displaystyle w=\lambda v}$ ${\displaystyle \lambda \in \mathbb {C} }$

Generic matrices

Generic elements in algebraic groups

An element is called generic if its Bruhat decomposition of the form ${\displaystyle g\in G}$

${\displaystyle g=b_{1}w_{0}b_{2}}$

with any and is the longest element in the Weyl group . ${\displaystyle b_{1},b_{2}\in B}$ ${\displaystyle w_{0}}$ ${\displaystyle W}$

Generic elements in GL (n, C)

A (real or complex) matrix is generic if the condition for all of the minors ${\displaystyle n\times n}$ ${\displaystyle A=(a_{ij})_{i,j=1,\ldots ,n}}$ ${\displaystyle k=1,\ldots ,n}$ ${\displaystyle \det \left(A_{\left\{k,\ldots ,n\right\},\left\{1,\ldots ,n-k+1\right\}}\right)}$

${\displaystyle \det \left(A_{\left\{k,\ldots ,n\right\},\left\{1,\ldots ,n-k+1\right\}}\right)\neq 0}$

fulfill.

Normal form of generic matrices

Each generic matrix can be uniquely identified as a ${\displaystyle A\in GL(n,\mathbb {C} )}$

${\displaystyle A=xqy^{-1}}$

decompose with upper triangular matrices and an antidiagonal matrix. The entries from and are given by ${\displaystyle x,y}$ ${\displaystyle q}$ ${\displaystyle x,y}$ ${\displaystyle q}$

${\displaystyle q_{n,1}=a_{n,1}}$

${\displaystyle q_{n-j+1,j}=(-1)^{j-1}{\frac {\det(A_{\left\{n-j+1,\ldots ,n\right\}\left\{1,\ldots ,j\right\}})}{\det(A_{\left\{n-j+2,\ldots ,n\right\}\left\{1\,\ldots ,j-1\right\}})}}}$     For ${\displaystyle 1<j\leq n}$

${\displaystyle x_{ij}={\frac {\det A_{\left\{i,j+1,\ldots ,n\right\}\left\{1\,\ldots ,n-j+1\right\}})}{\det A_{\left\{j\,\ldots ,n\right\}\left\{1\,\ldots ,n-j+1\right\}})}}}$     For ${\displaystyle j>i}$

${\displaystyle y_{ij}=(-1)^{i+j}{\frac {\det(A_{\{n-j+2\,\ldots ,n\}\{1\,\ldots ,{\hat {i}},\ldots ,j\}})}{\det(A_{\left\{n-j+2\,\ldots ,n\right\}\left\{1\,\ldots ,j-1\right\}})}}}$     for , ${\displaystyle j>i}$

where the hat notation stands for the deletion of the -th row or column. ${\displaystyle {\hat {i}}}$ ${\displaystyle i}$

literature

• Armand Borel . Linear Algebraic Groups (2nd ed.). New York: Springer-Verlag, 1991. ISBN 0-387-97370-2 .
• Nicolas Bourbaki , Lie Groups and Lie Algebras: Chapters 4-6 (Elements of Mathematics) , Springer-Verlag, 2008. ISBN 3-540-42650-7

Web links

• George Lusztig : Bruhat decomposition and applications (overview of the history and applications of Bruhat decomposition)
• Bruhat decomposition (nLab)

Individual evidence

1. ↑ Chapter 9 in: S. Garoufalidis, D. Thurston, C. Zickert, The complex volume of SL (n, C) -representations of 3-manifolds, Duke Math. J., Volume 164, Number 11 (2015), 2099 -2160. online (ArXiv)