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In mathematics, the '''Bruhat decomposition''' (named after [[François Bruhat]]) G = BWB into cells can be regarded as a general expression of the principle of [[Gauss–Jordan elimination]], which generically writes a matrix as a product of an upper triangular and lower triangular matrices—but with exceptional cases. It is related to the [[Schubert cell]] decomposition of Grassmannians: see [[Weyl group]] for this.
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


More generally, any group with a [[(B,N) pair]] has a Bruhat decomposition.
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==Definitions==
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*''G'' is a [[connected space|connected]], [[reductive group|reductive]] [[algebraic group]] over an [[algebraically closed field]].
*''B'' is a [[Borel subgroup]] of ''G''
*''W'' is a [[Weyl group]] of ''G'' corresponding to a maximal torus of ''B''.


The '''Bruhat decomposition''' of ''G'' is the decomposition
'''MathML'''
:<math>G=BWB =\coprod_{w\in W}BwB</math>
:<math forcemathmode="mathml">E=mc^2</math>
of ''G'' as a disjoint union of [[double coset]]s of ''B'' parameterized by the elements of the Weyl group ''W''. (Note that although ''W'' is not in general a subgroup of ''G'', the coset ''wB'' is still well defined.)


== Examples ==
<!--'''PNG''' (currently default in production)
Let ''G'' be the [[general linear group]] '''GL'''<sub>n</sub> of invertible <math>n \times n</math> matrices with entries in some algebraically closed field, which is a reductive group. Then the Weyl group ''W'' is isomorphic to the [[symmetric group]] ''S<sub>n</sub>'' on ''n'' letters, with [[permutation matrices]] as representatives. In this case, we can take ''B'' to be the subgroup of upper triangular invertible matrices, so Bruhat decomposition says that one can write any invertible matrix ''A'' as a product ''U<sub>1</sub>PU<sub>2</sub>'' where ''U<sub>1</sub>'' and ''U<sub>2</sub>'' are upper triangular, and ''P'' is a permutation matrix. Writing this as ''P = U<sub>1</sub><sup>-1</sup>AU<sub>2</sub><sup>-1</sup>'', this says that any invertible matrix can be transformed into a permutation matrix via a series of row and column operations, where we are only allowed to add row ''i'' (resp. column ''i'') to row ''j'' (resp. column ''j'') if ''i>j'' (resp. ''i<j''). The row operations correspond to ''U<sub>1</sub><sup>-1</sup>'', and the column operations correspond to ''U<sub>2</sub><sup>-1</sup>''.
:<math forcemathmode="png">E=mc^2</math>


The [[special linear group]] '''SL'''<sub>n</sub> of invertible <math>n \times n</math> matrices with [[determinant]] 1 is a [[semisimple algebraic group|semisimple group]], and hence reductive. In this case, ''W'' is still isomorphic to the symmetric group ''S<sub>n</sub>''. However, the determinant of a permutation matrix is the sign of the permutation, so to represent an odd permutation in '''SL'''<sub>n</sub>, we can take one of the nonzero elements to be -1 instead of 1. Here ''B'' is the subgroup of upper triangular matrices with determinant 1, so the interpretation of Bruhat decomposition in this case is similar to the case of '''GL'''<sub>n</sub>.
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


== Geometry ==
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].
The cells in the Bruhat decomposition correspond to the [[Schubert cell]] decomposition of Grassmannians. The dimension of the cells corresponds to the [[length function|length]] of the word ''w'' in the Weyl group. [[Poincaré duality]] constrains the topology of the cell decomposition, and thus the algebra of the Weyl group; for instance, the top dimensional cell is unique (it represents the [[fundamental class]]), and corresponds to the [[longest element of a Coxeter group]].


==Computations==
==Demos==
The number of cells in a given dimension of the Bruhat decomposition are the coefficients of the ''q''-polynomial<ref>[http://math.ucr.edu/home/baez/week186.html This Week's Finds in Mathematical Physics, Week 186]</ref> of the associated [[Dynkin diagram]].


==See also==
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:
* [[Lie group decompositions]]
* [[Birkhoff factorization]], a special case of the Bruhat decomposition for affine groups.


==Notes==
<references/>


==References==
* accessibility:
*[[Armand Borel|Borel, Armand]]. Linear Algebraic Groups (2nd ed.). New York: Springer-Verlag. ISBN 0-387-97370-2.
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
*[[Nicolas Bourbaki|Bourbaki, Nicolas]], ''Lie Groups and Lie Algebras: Chapters 4-6 (Elements of Mathematics)'', ISBN 3-540-42650-7
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


[[Category:Lie groups]]
==Test pages ==
[[Category:algebraic groups]]


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==Bug reporting==
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

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Bug reporting

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