flag

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

$$V_1\subset V_2\subset \mathrm{\cdots }\subset V_n=V$$

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, ifthe first $d_1$ vectors give a basis of $V_1$, the first $d_2$ vectorsgive a basis of $V_2$, etc. Thus, an alternate characterization of acomplete flag, is that the first $k$ elements of an adapted basis area basis of $V_k$.

Example

Let us consider ${ℝ}^n$. For each $k=1,\mathrm{\dots },n$ let $V_k$ be thespan of $e_1,\mathrm{\dots },e_k$, where $e_j$ denotes the $j^{\text{th}}$ basicvector, i.e. the column vector with $1$ in the $j^{\text{th}}$ position andzeros everywhere else. The $V_k$ give a complete flag in ${ℝ}^n$ .The list $(e_1,e_2,\mathrm{\dots },e_n)$ is an adapted basis relative to thisflag, 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.