Gritsenko, Skoruppa and Zagier associated to a root system $R$ theta block $\vartheta_R$, which is Jacobi form of lattice index. We classify the blocks $\vartheta_R$ $q$-order $1$ show that their Gritsenko lift strongly-reflective Borcherds product singular weight, related Conway's group $\operatorname{Co}_0$. As corollary we obtain proof conjecture by Poor Yuen for pure obtained as specializat...

A root system is splint if it a decomposition into union of two disjoint systems. Examples such systems arise naturally in studying embeddings reductive Lie subalgebras simple algebras. Given system, one can try to understand its branching rule. In this paper we discuss methods rules, and give precise formulas for specific cases, including the restriction functor from exceptional algebra $\math...

We tackle several problems related to a finite irreducible crystallographic root system $\Phi$ in the real vector space $\mathbb E$. In particular, we study combinatorial structure of subsets cut by affine subspaces E$ and their projections. As byproducts, obtain easy algebraic proofs refinements Oshima's Lemma result Kostant, partial towards resolution problem Hopkins Postnikov, new enumerativ...