nLab flag

Definition

General

In geometry, a flag is a chain of incidence relations, as for example between distinct linear subspaces

$$V_0 \subseteq V_1 \subseteq \ldots \subseteq V_n$$

of a fixed vector space $V$, or between isotropic subspaces, etc. A flag is complete if $\dim V_i = i$ for each $i \in \{0, \ldots, n\}$.

Flags of posets

Generally speaking, if $P$ is a poset, a flag is a chain

$$x_0 \lt x_1 \lt \ldots \lt x_n$$

and the set of elements $\{x_0, x_1, \ldots, x_n\}$ can be thought of as an $n$-simplex of a simplicial complex whose vertices are the poset elements. Hence we have a functor

$$Flag: Pos \to SimpComplex$$

In the other direction, there is an underlying functor

$$U: SimpComplex \to Pos$$

which sends a simplicial complex $(V, \Sigma)$ to $\Sigma$ (regarded as a poset ordered by inclusion). The composite

$$Flag \circ U: SimpComplex \to SimpComplex$$

is called the subdivision functor, or, more exactly, the barycentric subdivision functor.

Related concepts

• flag variety