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''of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f0405502.png" /> in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f0405503.png" />-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f0405504.png" />''
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''of type $  \nu $
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in an $  n $ -
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dimensional vector space $  \nu $ ''
  
A collection of linear subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f0405505.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f0405506.png" /> of corresponding dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f0405507.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f0405508.png" /> (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f0405509.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f04055010.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f04055011.png" />). A flag of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f04055012.png" /> is called a complete flag or a full flag. Any two flags of the same type can be mapped to each other by some linear transformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f04055013.png" />, that is, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f04055014.png" /> of all flags of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f04055015.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f04055016.png" /> is a homogeneous space of the general linear group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f04055017.png" />. The unimodular group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f04055018.png" /> also acts transitively on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f04055019.png" />. Here the stationary subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f04055020.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f04055021.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f04055022.png" /> (and also in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f04055023.png" />) is a parabolic subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f04055024.png" /> (respectively, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f04055025.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f04055026.png" /> is a complete flag in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f04055027.png" />, defined by subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f04055028.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f04055029.png" /> is a complete triangular subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f04055030.png" /> (respectively, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f04055031.png" />) relative to a basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f04055032.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f04055033.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f04055034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f04055035.png" />. In general, quotient spaces of linear algebraic groups by parabolic subgroups are sometimes called flag varieties. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f04055036.png" />, a flag of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f04055037.png" /> is simply an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f04055038.png" />-dimensional linear subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f04055039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f04055040.png" /> is the [[Grassmann manifold|Grassmann manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f04055041.png" />. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f04055042.png" /> is the projective space associated with the vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f04055043.png" />. Every flag variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f04055044.png" /> can be canonically equipped with the structure of a projective algebraic variety (see [1]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f04055045.png" /> is a real or complex vector space, then all the varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f04055046.png" /> are compact. Cellular decompositions and cohomology rings of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040550/f04055047.png" /> are known (see [3], and also [[Bruhat decomposition|Bruhat decomposition]]).
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A collection of linear subspaces $  V $
 +
of $  V _{1} \dots V _{k} $
 +
of corresponding dimensions $  V $ ,  
 +
such that $  n _{1} \dots n _{k} $ (
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here $  V _{1} \subset \dots \subset V _{k} $ ,
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$  \nu = (n _{1} \dots n _{k} ) $ ;
 +
$  1 \leq k \leq n - 1 ;\quad 0 < n _{1} < \dots < n _{k} < n $ ).  
 +
A flag of type $  \nu _{0} = (1 \dots n - 1 ) $
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is called a complete flag or a full flag. Any two flags of the same type can be mapped to each other by some linear transformation of $  V $ ,  
 +
that is, the set $  F _ \nu  (V) $
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of all flags of type $  \nu $
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in $  V $
 +
is a homogeneous space of the general linear group $  \mathop{\rm GL}\nolimits (V) $ .  
 +
The unimodular group $  \mathop{\rm SL}\nolimits (V) $
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also acts transitively on $  F _ \nu  (V) $ .  
 +
Here the stationary subgroup $  H _{F} $
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of $  F $
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in $  \mathop{\rm GL}\nolimits (V) $ (
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and also in $  \mathop{\rm SL}\nolimits (V) $ )  
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is a parabolic subgroup of $  \mathop{\rm GL}\nolimits (V) $ (
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respectively, of $  \mathop{\rm SL}\nolimits (V) $ ).  
 +
If $  F $
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is a complete flag in $  V $ ,  
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defined by subspaces $  V _{1} \subset \dots \subset V _ {n - 1} $ ,  
 +
then $  H _{F} $
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is a complete triangular subgroup of $  \mathop{\rm GL}\nolimits (V) $ (
 +
respectively, of $  \mathop{\rm SL}\nolimits (V) $ )  
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relative to a basis $  e _{1} \dots e _{n} $
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of $  V $
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such that $  e _{i} \in V _{i} $ ,  
 +
$  i = 1 \dots n $ .  
 +
In general, quotient spaces of linear algebraic groups by parabolic subgroups are sometimes called flag varieties. For $  k = 1 $ ,  
 +
a flag of type $  (n _{1} ) $
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is simply an $  n _{1} $ -
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dimensional linear subspace of $  V $
 +
and $  F _ {(n _{1} )} (V) $
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is the [[Grassmann manifold|Grassmann manifold]] $  G _ {n, n _{1}} = \mathop{\rm Gr}\nolimits _ {n _{1}} (V) $ .  
 +
In particular, $  F _{(1)} (V) $
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is the projective space associated with the vector space $  V $ .  
 +
Every flag variety $  F _ \nu  (V) $
 +
can be canonically equipped with the structure of a projective algebraic variety (see ). If $  V $
 +
is a real or complex vector space, then all the varieties $  F _ \nu  (V) $
 +
are compact. Cellular decompositions and cohomology rings of the $  F _ \nu  (V) $
 +
are known (see , and also [[Bruhat decomposition|Bruhat decomposition]]).
  
 
For references see [[Flag structure|Flag structure]].
 
For references see [[Flag structure|Flag structure]].

Latest revision as of 13:30, 15 December 2019

of type $ \nu $ in an $ n $ - dimensional vector space $ \nu $


A collection of linear subspaces $ V $ of $ V _{1} \dots V _{k} $ of corresponding dimensions $ V $ , such that $ n _{1} \dots n _{k} $ ( here $ V _{1} \subset \dots \subset V _{k} $ , $ \nu = (n _{1} \dots n _{k} ) $ ; $ 1 \leq k \leq n - 1 ;\quad 0 < n _{1} < \dots < n _{k} < n $ ). A flag of type $ \nu _{0} = (1 \dots n - 1 ) $ is called a complete flag or a full flag. Any two flags of the same type can be mapped to each other by some linear transformation of $ V $ , that is, the set $ F _ \nu (V) $ of all flags of type $ \nu $ in $ V $ is a homogeneous space of the general linear group $ \mathop{\rm GL}\nolimits (V) $ . The unimodular group $ \mathop{\rm SL}\nolimits (V) $ also acts transitively on $ F _ \nu (V) $ . Here the stationary subgroup $ H _{F} $ of $ F $ in $ \mathop{\rm GL}\nolimits (V) $ ( and also in $ \mathop{\rm SL}\nolimits (V) $ ) is a parabolic subgroup of $ \mathop{\rm GL}\nolimits (V) $ ( respectively, of $ \mathop{\rm SL}\nolimits (V) $ ). If $ F $ is a complete flag in $ V $ , defined by subspaces $ V _{1} \subset \dots \subset V _ {n - 1} $ , then $ H _{F} $ is a complete triangular subgroup of $ \mathop{\rm GL}\nolimits (V) $ ( respectively, of $ \mathop{\rm SL}\nolimits (V) $ ) relative to a basis $ e _{1} \dots e _{n} $ of $ V $ such that $ e _{i} \in V _{i} $ , $ i = 1 \dots n $ . In general, quotient spaces of linear algebraic groups by parabolic subgroups are sometimes called flag varieties. For $ k = 1 $ , a flag of type $ (n _{1} ) $ is simply an $ n _{1} $ - dimensional linear subspace of $ V $ and $ F _ {(n _{1} )} (V) $ is the Grassmann manifold $ G _ {n, n _{1}} = \mathop{\rm Gr}\nolimits _ {n _{1}} (V) $ . In particular, $ F _{(1)} (V) $ is the projective space associated with the vector space $ V $ . Every flag variety $ F _ \nu (V) $ can be canonically equipped with the structure of a projective algebraic variety (see ). If $ V $ is a real or complex vector space, then all the varieties $ F _ \nu (V) $ are compact. Cellular decompositions and cohomology rings of the $ F _ \nu (V) $ are known (see , and also Bruhat decomposition).

For references see Flag structure.

How to Cite This Entry:
Flag. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Flag&oldid=43107
This article was adapted from an original article by D.V. Alekseevskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article