Combinatorial Curve Neighborhood of the Affine Flag Manifold of Type $A_{n-1}^1$

Abstract

Let $\mathscr{X}$ be the affine flag manifold of Lie type $A_{n-1}^{(1)}$ where $n \geq 3$ and let $W_{\text{aff}}$ be the associated affine Weyl group. The moment graph for $\mathscr{X}$ encodes the torus fixed points (corresponding to elements of the affine Weyl group $W_{\text{aff}}$) and the torus stable curves in $\mathscr{X}$. Given a fixed point $u\in W_{\text{aff}}$ and a degree $\mathbf{d}=(d_0,d_1,...,d_{n-1})\in \mathbb{Z}_{\geq 0}^{n}$, the combinatorial curve neighborhood is the set of maximal elements in the moment graph of $\mathscr{X}$ which can be reached from $u'\leq u$ by a chain of curves of total degree $\leq \mathbf{d}$. In this paper we give combinatorial formulas and algorithms for calculating these elements in $\mathscr{X}$.