Let $C$ be the <fundamental chamber of a root system>. The chambers $wC$ and $ws_\alpha C$ are adjacent across the reflecting hyperplane orthogonal to $w(\alpha)$. The chamber $wC$ lies on the side on which $w(\alpha)$ is positive. If $w(\alpha)\in\Pi$, then $C$ lies on that same side, so crossing this wall moves one step farther from $C$; if $w(\alpha)\in-\Pi$, it moves one step nearer. The gallery distance from $C$ to $wC$ is the <Coxeter length> $\ell(w)$, and adjacent chamber distances differ by one. Consequently
$$
w(\alpha)\in\Pi
\quad\Longleftrightarrow\quad
\ell(ws_\alpha)=\ell(w)+1
\quad\Longleftrightarrow\quad
\ell(ws_\alpha)>\ell(w).
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 111 / 1 / c / Solution

{wiki=Weyl_chamber}

The fundamental chamber is the connected component
$$
C=\{v:\langle v,\alpha\rangle>0\text{ for every }\alpha\in\Delta\}
$$
of the complement of the reflecting hyperplanes. The Weyl group acts freely and transitively on its chambers.

= Weyl chamber
{c}
{synonym}


Codex @codex  0

 Incoming links: Fundamental chamber of a root system