Vol. 10, No. 2, 2017

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Combinatorial curve neighborhoods for the affine flag manifold of type $A_1^1$

Leonardo C. Mihalcea and Trevor Norton

Vol. 10 (2017), No. 2, 317–325
Abstract

Let $X$ be the affine flag manifold of Lie type ${A}_{1}^{1}$. Its moment graph encodes the torus fixed points (which are elements of the infinite dihedral group ${D}_{\infty }$) and the torus stable curves in $X$. Given a fixed point $u\in {D}_{\infty }$ and a degree $d=\left({d}_{0},{d}_{1}\right)\in {ℤ}_{\ge 0}^{2}$, the combinatorial curve neighborhood is the set of maximal elements in the moment graph of $X$ which can be reached from $u$ using a chain of curves of total degree $\le d$. In this paper we give a formula for these elements, using combinatorics of the affine root system of type ${A}_{1}^{1}$.

Keywords
affine flag manifolds, moment graph, curve neighborhood
Mathematical Subject Classification 2010
Primary: 05E15
Secondary: 17B67, 14M15