# Horospherical varieties on Grassmannian.info

Last week I talked about how the Hodge diamond cutter now deals with horospherical varieties of rank 1. Now I've also implemented their properties for Grassmannian.info, see the overview table. Even though they are not homogeneous varieties, their geometry is so close to them that it makes sense to include them.

A *horospherical* variety $X$ is a variety with an action by a reductive group $G$, such that $X$ has a dense $B$-orbit (where $B$ is a Borel subgroup), and if we let $H$ denote the stabiliser of a point in this dense orbit, then $H$ contains a conjugate of the maximal unipotent subgroup of $G$ inside $B$. In other words: we don't let $B$ act completely transitively, but

There exists an interesting geometric classification in the case of rank 1 by Pasquier, and this classification together with some of the geometric properties of these varieties is now available. I'm still working on the quantum spectrum, and I hope I haven't made any silly mistakes. Let me know what you think in the comments below, and don't forget to like and subscribe! Oh wait, this isn't YouTube...