3 problems

Let $\mathcal{L}=\mathbb{Z}^{a-1}$, let $\operatorname{Box}$ be the fundamental box, and for coprime $a,c$ let $c'$ be the largest integer less than $c$ with $\gcd(a,c')=1$. A stan…

Let $\mathcal{L}=\mathbb{Z}^{a-1}$ be the weight lattice, let $\mathcal{R}$ be its root sublattice, and let $\mathsf{T}(x_1,\ldots,x_{a-1})=x_1+2x_2+\cdots+(a-1)x_{a-1}$. A Johnson…

Let $\mathcal{L}=\mathbb{Z}^{a-1}$ be the weight lattice, let … be the root lattice, and let $\mathsf{T}(x_1,\ldots,x_{a-1})=x_1+2x_2+\cdots+(a-1)x_{a-1}$. Johnson's conjecture. Th…