Combinatorial curve neighborhoods for the affine flag manifold of type A11

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 $d = (d_{0}, d_{1}) \in ℤ_{\geq 0}^{2}$ , 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 $\leq d$ . 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

Milestones

Received: 13 December 2015

Accepted: 1 April 2016

Published: 10 November 2016

Communicated by Jim Haglund

Authors

Leonardo C. Mihalcea
Department of Mathematics Virginia Tech University Blacksburg, VA 24061 %-0123 United States

Trevor Norton
Department of Mathematics Virginia Tech University Blacksburg, VA 24061 United States

Vol. 10, No. 2, 2017

Leonardo C. Mihalcea and Trevor Norton

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors