Vector Space Flag

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

 V_0 subset V_1 subset ... subset V_r,

where all inclusions are strict. Hence


so that r<=n. If equality holds, then dimV_i=i for all i, and the flag is called complete or full. In this case it is a composition series of V.

A full flag can be constructed by fixing a basis v_1,...,v_n of V, and then taking V_i=<v_1,...,v_i> for all i=0,...,n.

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 R^3

 {origin} subset {xy-plane} subset R^3

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

 {origin} subset {x-axis} subset {xy-plane} subset R^3


 {origin} subset {y-axis} subset {xy-plane} subset R^3.

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.

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