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# Vector Space Flag

An ascending chain of subspaces of a vector space. If is an -dimensional vector space, a flag of is a filtration

 (1)

where all inclusions are strict. Hence

 (2)

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)

can be completed by switching in any line of the -plane passing through the origin. Two different full flags are, for example,

 (4)

and

 (5)

Schubert varieties are projective varieties defined from flags.

This entry contributed by Margherita Barile

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## Cite this as:

Barile, Margherita. "Vector Space Flag." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/VectorSpaceFlag.html