An ascending chain of subspaces of a vector space. If is an -dimensional vector space, a flag of is a filtration
(1)
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where all inclusions are strict. Hence
(2)
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so that . If equality holds, then for all , and the flag is called complete or full. In this case it is a composition series of .
A full flag can be constructed by fixing a basis of , and then taking for all .
A flag of any length can be obtained from a full flag by taking out some of the subspaces. Conversely, every flag can be completed to a full flag by inserting suitable subspaces. In general, this can be done in different ways. The following flag of
(3)
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can be completed by switching in any line of the -plane passing through the origin. Two different full flags are, for example,
(4)
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and
(5)
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Schubert varieties are projective varieties defined from flags.