Balázs Elek

A Kazhdan-Lusztig atlas, introduced by He, Knutson and Lu, on a stratified variety $(V,\mathcal{Y})$ is a way of modeling the stratification $\mathcal{Y}$ of $V$ locally using the stratification of Kazhdan-Lusztig varieties $X^{w}_o\cap X_{v}$. We are interested in classifying smooth toric surfaces with Kazhdan-Lusztig atlases. This involves finding a degeneration of $V$ to a union of Richardson varieties in the flag variety $H/B_H$ of some Kac-Moody group $H$. We determine which toric surfaces have a chance at having a Kazhdan-Lusztig atlas by looking at their moment polytopes, then describe a way to find a suitable group $H$. More precisely, we find that (up to equivalence) there are $19$ or $20$ broken toric surfaces admitting simply-laced atlases, and that there are at most $7543$ broken toric surfaces where $H$ is any Kac-Moody group.