2020/10/12 by Chapelier-Laget, Nathan
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2010.05602
Let Wa be an affine Weyl group with corresponding finite root system Φ. In \citeJYS1 Jian-Yi Shi characterized each element w ∈ Wa by a Φ+-tuple of integers (k(w,α))α∈ Φ+ subject to certain conditions. In \citeNC1 a new interpretation of the coefficients k(w,α) is given. This description led us to define an affine variety \widehatXWa, called the Shi variety of Wa, whose integral points are in bijection with Wa. It turns out that this variety has more than one irreducible component, and the set of these components, denoted H0(\widehatXWa), admits many interesting properties. In particular the group Wa acts on it. In this article we show that the set of irreducible components of \widehatX_W(\widetildeAn) is in bijection with the conjugacy class of (1~2~⋯~n+1) ∈ W(An) = Sn+1. We also compute the action of W(An) on H0(\widehatX_W(\widetildeAn)).