The Combinatorial Curve Neighborhoods of Affine Flag Manifold in Type A_{n-1}^(1)

Abstract

Let X be the affine flag manifold of Lie type A_{n-1}^(1) where n ≥ 3 and let W_{aff} be the associated affine Weyl group. The moment graph for X encodes the torus fixed points (which are elements of the affine Weyl group W_{aff} and the torus stable curves in X. Given a fixed point u ∈ Waff and a degree d = (d_0, d_1, ..., d_{n−1}) ∈ Z^{n≥0}, the combinatorial curve neighborhood is the set of maximal elements in the moment graph of X which can be reached from u′ ≤ u by a chain of curves of total degree ≤ d. In this thesis we give combinatorial formulas and algorithms for calculating these elements.