Flags in A Vector Space
A flag in a finite dimensional vector space V over a field F is an increasing sequence of subspaces, where "increasing" means each is a proper subspace of the next (see filtration):
If we write the dim Vi = di then we have
where n is the dimension of V. Hence, we must have k ≤ n. A flag is called a complete flag if di = i, otherwise it is called a partial flag. The signature of the flag is the sequence (d1, … dk).
A partial flag can be obtained from a complete flag by deleting some of the subspaces. Conversely, any partial flag can be completed (in many different ways) by inserting suitable subspaces.
Read more about this topic: Generalized Flag Variety
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—Blaise Pascal (1623–1662)