# flag

Let $V$ be a finite-dimensional vector space^{}. A filtration^{} of
subspaces^{}

$${V}_{1}\subset {V}_{2}\subset \mathrm{\cdots}\subset {V}_{n}=V$$ |

is called a *flag* in $V$.
We speak of a complete flag when

$$dim{V}_{i}=i$$ |

for each $i=1,\mathrm{\dots},n$.

Next, putting

$${d}_{k}=dim{V}_{k},k=1,\mathrm{\dots}n,$$ |

we say that a list of vectors
$({u}_{1},\mathrm{\dots},{u}_{{d}_{n}})$ is an *adapted basis* relative to the flag, if
the first ${d}_{1}$ vectors give a basis of ${V}_{1}$, the first ${d}_{2}$ vectors
give a basis of ${V}_{2}$, etc. Thus, an alternate characterization of a
complete flag, is that the first $k$ elements of an adapted basis are
a basis of ${V}_{k}$.

## Example

Let us consider ${\mathbb{R}}^{n}$. For each $k=1,\mathrm{\dots},n$ let ${V}_{k}$ be the
span of ${e}_{1},\mathrm{\dots},{e}_{k}$, where ${e}_{j}$ denotes the ${j}^{\text{th}}$ basic
vector, i.e. the column vector^{} with $1$ in the ${j}^{\text{th}}$ position and
zeros everywhere else. The ${V}_{k}$ give a complete flag in ${\mathbb{R}}^{n}$ .
The list $({e}_{1},{e}_{2},\mathrm{\dots},{e}_{n})$ is an adapted basis relative to this
flag, but the list $({e}_{2},{e}_{1},\mathrm{\dots},{e}_{n})$ is not.

## Generalizations.

More generally, a flag can be defined as a maximal chain in a partially ordered set^{}. If one considers the poset consisting of subspaces of a (finite dimensional) vector space, one recovers the definition given above.

Title | flag |
---|---|

Canonical name | Flag |

Date of creation | 2013-03-22 12:42:35 |

Last modified on | 2013-03-22 12:42:35 |

Owner | rmilson (146) |

Last modified by | rmilson (146) |

Numerical id | 9 |

Author | rmilson (146) |

Entry type | Definition |

Classification | msc 06A06 |

Classification | msc 15A03 |

Defines | adapted basis |

Defines | complete flag |