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My favorite answer to #2 and #3 is Kostant's "Find the highest root game", which is written up in detail in section 5.4 of Balázs Elek's notes on reflection groups. It is not hard to show that all pla …

In general if $T$ acts on a projective variety $X$ with moment polytope $\Phi(X)$, then a general point $x\in X$ will have $\Phi(\overline{T\cdot x}) = \Phi(X)$ i.e. be an abnormal toric variety with …

I apologize for asking questions that seem likely to be answered in Conway & Sloane's "Sphere Packings, Lattices, and Groups" if I knew where to look. Let $L$ be the unique* even unimodular lattice o …

This is only a slight modification of the argument already given, but I liked it enough to type it in. Since $W$ acts with no stabilizer on the open Weyl chamber $K$, for $K$ to contain a root $\beta …

Using the correspondences $K/Ad\ K \cong T/W \cong (\mathfrak t/Q^\vee)/W \cong \mathfrak t/(Q^\vee \rtimes W) = \mathfrak t/\hat W =: A$, you can think about the Weyl alcove $A$ as parametrizing the …

I like being able to say "the highest root is always long".

The faces are all of the following form: $w W_P / Stab_W(\xi)$, where $W_P$ varies over the subgroups generated by subsets of the simple reflections. In particular, for $\xi$ regular, the number of th …

The mnemonic I use: if the diagram has a natural involution, then $-w_0$ induces it, otherwise $w_0 = -1$. The only place this fails is in $D_n$, where one can switch the antlers, but shouldn't always …