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

TL;DR

This paper develops combinatorial formulas and algorithms to compute the combinatorial curve neighborhoods in the affine flag manifold of type A, capturing the structure of torus fixed points and stable curves.

Contribution

It introduces explicit combinatorial methods for calculating curve neighborhoods in the affine flag manifold of type A, advancing understanding of its geometric and combinatorial structure.

Findings

Provides formulas for combinatorial curve neighborhoods.

Develops algorithms for their computation.

Enhances understanding of the moment graph structure.

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 $\mathscr{X}$.