Weight-space and dimension conjecture for affine nilCoxeter algebra simple modules

Let $A$) be a nilCoxeter algebra of type $\tilde A_{n-1}$ with $n\geq 2$. Work over an algebraically closed field of characteristic zero. A weight space of a simple $A$-module is the common eigenspace associated with a character of the commutative weight subalgebra described in the paper. For a weight whose stabiliser is a conjugate of the Young subgroup

$$\mathfrak{S}_{n_1}\times\dots\times\mathfrak{S}_{n_s},$$

with $n_1+\dots+n_s=n$, let the corresponding weight space be one of the weight spaces of the simple module.

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

$$\binom{n}{n_1\dots n_s}=\frac{n!}{n_1!\dots n_s!},$$

where the stabilisers of the corresponding weight spaces are the conjugates of the Young subgroup $\mathfrak{S}_{n_1}\times\dots\times\mathfrak{S}_{n_s}$.

The conjecture gives a uniform description of the simple modules for affine nilCoxeter algebras. It is verified in the cases of types $\tilde A_{n-1}$ for $n=2$, $3$, and $4$; the general case remains open.