Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 102 / 2

A Weyl chamber is a connected component ofA root basis $\Delta$ is a basis of $E$ made of roots such that every root is an integer combination of $\Delta$ whose nonzero coefficients all have one sign.

Choose a regular vector $\gamma$, meaning $(\gamma ,\alpha )\ne 0$ for every root. DeclareThe indecomposable roots in ${\Phi }_{\gamma }^{+}$ form a root basis ${\Delta }_{\gamma }$, and every root basis arises in this way. Its chamber is the component containing $\gamma$.

Solved by gpt-5.6-sol high.

Root bases correspond bijectively to Weyl chambers: the walls of a chamber determine its inward simple roots. The Weyl group acts transitively on the chambers. One proof joins interior points of two chambers by a generic line segment. Each time the segment crosses one reflecting hyperplane, reflect the remaining segment across that wall; the resulting product of root reflections sends the first chamber to the second. It consequently sends the first root basis to the second. Thus $W$ acts transitively on root bases.

Solved by gpt-5.6-sol high.

We induct on the Coxeter length $\ell (w)$. There is nothing to prove when $\ell (w)=0$. Otherwise choose a simple root $\alpha$ such thatequivalently, $w\alpha$ is negative. Since $\lambda$ and $\mu =w\lambda$ lie in the Closed dominant Weyl chamber,Therefore $⟨\lambda ,{\alpha }^{\vee }⟩=0$, and the simple reflection $s_{\alpha }$ fixes $\lambda$. Moreover,The induction hypothesis writes $w{s}_{\alpha }$ as a product of simple reflections that fix $\lambda$. Multiplying on the right by $s_{\alpha }$ gives the required expression for $w$. This is the Weyl stabilizer of a dominant point lemma.

Solved by gpt-5.6-sol high.

For existence, choose $w\in W$ maximizing $(w\lambda ,\rho )$, where $\rho$ lies in the interior of the dominant chamber. If $⟨w\lambda ,{\alpha }^{\vee }⟩<0$ for a simple root $\alpha$, thencontradicting maximality. Hence $w\lambda$ is dominant.

For uniqueness, suppose $\lambda$ and $\mu$ are dominant and $\mu =w\lambda$. Part (c) writes $w$ as a product of simple reflections fixing $\lambda$, so $\mu =\lambda$. Every Weyl orbit therefore has exactly one representative in the closed dominant chamber.

Solved by gpt-5.6-sol high.