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## User: nathan-chapelier-laget

### Title: Shi variety corresponding to an affine Weyl group

In this article we show that there exists a bijection between any affine Weyl
group $W_a$ and the integral points of an affine variety, denoted
$\hat{X}_{W_a}$, which we call the Shi variety of $W_a$. In order to do so, we
use Jian-Yi Shi's characterization of alcoves in affine Weyl groups. We then
study this variety further. We highlight combinatorial properties of
irreducible components of $\hat{X}_{W_a}$ and we show how they are related to a
fundamental set $P_{\mathcal{H}}$ endowed with the right weak order of W_a. In
the last section we exhibit a link between the first cohomology group of the
underlying crystallographic group and some components of $\hat{X}_{W_a}$.

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