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Flag (linear algebra) ：ウィキペディア英語版

Flag (linear algebra)

In mathematics, particularly in linear algebra, a flag is an increasing sequence of subspaces of a finite-dimensional vector space ''V''. Here "increasing" means each is a proper subspace of the next (see filtration):
:

\ = V_0 \sub V_1 \sub V_2 \sub \cdots \sub V_k = V.

If we write the dim ''V''_''i'' = ''d''_''i'' then we have
:

0 = d_0 < d_1 < d_2 < \cdots < d_k = n,

where ''n'' is the dimension of ''V'' (assumed to be finite-dimensional). Hence, we must have ''k'' ≤ ''n''. A flag is called a complete flag if ''d''_''i'' = ''i'', otherwise it is called a partial flag.
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.
The signature of the flag is the sequence (''d''₁, … ''d''_''k'').
Under certain conditions the resulting sequence resembles a flag with a point connected to a line connected to a surface.
==Bases==
An ordered basis for ''V'' is said to be adapted to a flag if the first ''d''_''i'' basis vectors form a basis for ''V''_''i'' for each 0 ≤ ''i'' ≤ ''k''. Standard arguments from linear algebra can show that any flag has an adapted basis.
Any ordered basis gives rise to a complete flag by letting the ''V''_''i'' be the span of the first ''i'' basis vectors. For example, the in R^''n'' is induced from the standard basis (''e''₁, ..., ''e''_''n'') where ''e''_''i'' denotes the vector with a 1 in the ''i''th slot and 0's elsewhere. Concretely, the standard flag is the subspaces:
:

0 < \left\langle e_1\right\rangle < \left\langle e_1,e_2\right\rangle < \cdots < \left\langle e_1,\ldots,e_n \right\rangle = K^n.

An adapted basis is almost never unique (trivial counterexamples); see below.
A complete flag on an inner product space has an essentially unique orthonormal basis: it is unique up to multiplying each vector by a unit (scalar of unit length, like 1, -1, ''i''). This is easiest to prove inductively, by noting that

v_i \in V_^\perp < V_i

, which defines it uniquely up to unit.
More abstractly, it is unique up to an action of the maximal torus: the flag corresponds to the Borel group, and the inner product corresponds to the maximal compact subgroup.〔Harris, Joe (1991). ''Representation Theory: A First Course'', p. 95. Springer. ISBN 0387974954.〕

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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