3 problems

Weight-space and dimension conjecture. The weight spaces of simple $A$-modules are one dimensional. Consequently, the dimension of a simple module is the multinomial coefficient

Let $\mathfrak{N}_m$ be the degree-$m$ piece of the nilCoxeter sequential 2-algebra, and let $\mathrm{Sym}^m(\mathbb{R}^2)$ denote the $m$-fold symmetric product of the plane. A ni…

Let $W$ be a Weyl group, let $R^+$ be its positive roots ordered by $\alpha\prec\beta$ when $\beta-\alpha$ is a positive sum of simple roots, and let $J(R^+,\prec)$ be the set of u…