Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 111 / 3 / d / Solution

If $W$ is finite, its finitely many reflecting hyperplanes divide $V^{\ast }$ into chambers, and the closures of these chambers are exactly the translates ${\sigma }^{\ast }(w)(D)$. Hence the Tits coneequals $V^{\ast }$.

Conversely, suppose $W$ is infinite and choose $f\in -C$, so $f(e_i)<0$ for every $i$. If $f\in U$, then $f\in {\sigma }^{\ast }(w)(D)$ for some $w$, and thereforefor every $i$. Since $f$ is strictly negative on every positive root and strictly positive on every negative root, each $\sigma (w)e_i$ must be negative. The length criterion givesfor every $i$. This contradicts the stated fact that in an infinite Coxeter group the length of every element can be increased by multiplication on the right by some simple generator. Thus $-C$ is not contained in $U$, and $U\ne V^{\ast }$.

Solved by gpt-5.6-sol high.