Flag 2002-06-01 23:55:00 2006-09-23 16:45:42 Definition adapted basis complete flag \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \newcommand{\reals}{\mathbb{R}} \newcommand{\natnums}{\mathbb{N}} \newcommand{\cnums}{\mathbb{C}} \newcommand{\znums}{\mathbb{Z}} \newcommand{\lp}{\left(} \newcommand{\rp}{\right)} \newcommand{\lb}{\left[} \newcommand{\rb}{\right]} \newcommand{\supth}{^{\text{th}}} \newtheorem{proposition}{Proposition} \newtheorem{definition}[proposition]{Definition} \newcommand{\nl}[1]{\PMlinkescapetext{{#1}}} \newcommand{\pln}[2]{\PMlinkname{#1}{#2}} 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 \emph{flag} in $V$. We speak of a {\em 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 \emph{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 $V_k$. \paragraph{Example} Let us consider $\reals^n$. For each $k=1,\ldots,n$ let $V_k$ be the span of $e_1,\ldots,e_k$, where $e_j$ denotes the $j\supth$ basic vector, i.e. the column vector with $1$ in the $j\supth$ position and zeros everywhere else. The $V_k$ give a complete flag in $\reals^n$ . 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. \paragraph{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.