# flag

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

 $V_{1}\subset V_{2}\subset\cdots\subset V_{n}=V$

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

 $\dim V_{i}=i$

for each $i=1,\ldots,n$.

Next, putting

 $d_{k}=\dim V_{k},\quad k=1,\ldots n,$

we say that a list of vectors $(u_{1},\ldots,u_{d_{n}})$ 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 $k=1,\ldots,n$ let $V_{k}$ be the span of $e_{1},\ldots,e_{k}$, 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 $(e_{1},e_{2},\ldots,e_{n})$ is an adapted basis relative to this flag, but the list $(e_{2},e_{1},\ldots,e_{n})$ 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 Flag 2013-03-22 12:42:35 2013-03-22 12:42:35 rmilson (146) rmilson (146) 9 rmilson (146) Definition msc 06A06 msc 15A03 adapted basis complete flag