## Spiral Schubert varieties for affine A2

##### Abstract

The geometry of Schubert varieties $X(w)$ for a Kac-Moody group $G$ is closely related to the corresponding affine Weyl group $W$.
A great deal of geometric information is encoded in the Bruhat order on $W$.
In particular, given a pair of elements $x le w$ in $W$, there are integers $q^w_x$ defined using the Bruhat order which can be used to determine rational smoothness of $X(w) $.
We prove general results relating the Bruhat order for $W$ of type $tilde A_2$ to the action of $W$ on $mathbb{R}^2, $ using the bijection of $W$ with the center points of the alcoves on $mathbb{R}^2$.
We apply these results to an interesting family of elements $w(ell)in W (ell in mathbb{N})$ called spiral elements. We show that $xle w(ell)$ if and only if the corresponding center point $xq$ lies in a region $R(ell)$ which is close to a triangle. Using this we determine all the $q^{w(ell)}_x$ and determine the set of
rationally smooth points of $X(w(ell))$. This leads to the proof of the lookup conjecture
for spiral Schubert varieties $X(w(ell))$.