A symmetric group action on the irreducible components of the Shi variety associated to W(A~n)

• Published in: Journal of algebraic combinatorics Vol. 58; no. 3; pp. 717 - 739
• Main Author: Chapelier-Laget, Nathan
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.11.2023; Springer Nature B.V; Springer Verlag
• Subjects: Combinatorics; Computer Science; Convex and Discrete Geometry; Group Theory and Generalizations; Lattices; Mathematics; Mathematics and Statistics; Order; Ordered Algebraic Structures; Irreducible components; Shi variety
• ISSN: 0925-9899; 1572-9192
• DOI: 10.1007/s10801-023-01243-5

Let W a be an affine Weyl group with corresponding finite root system Φ . In Shi (J Lond Math Soc (2) 35(1):42–55, 1987) characterized each element w ∈ W a by a Φ + -tuple of integers ( k ( w , α ) ) α ∈ Φ + subject to certain conditions. In Chapelier-Laget (Shi variety corresponding to an affine Weyl group. arXiv:2010.04310 , 2020) a new interpretation of the coefficients k ( w , α ) is given. This description led us to define an affine variety X ^ W a , called the Shi variety of W a , whose integral points are in bijection with W a . It turns out that this variety has more than one irreducible component, and the set of these components, denoted H 0 ( X ^ W a ) , admits many interesting properties. In particular the group W a acts on it. In this article we show that the set of irreducible components of X ^ W ( A ~ n ) is in bijection with the conjugacy class of ( 1 2 ⋯ n + 1 ) ∈ W ( A n ) = S n + 1 . We also compute the action of W ( A n ) on H 0 ( X ^ W ( A ~ n ) ) .