Combinatorial curve neighborhoods for the affine flag manifold of type A11

Mihalcea, Leonardo C.; Norton, Trevor

doi:10.2140/involve.2017.10.317

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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

DOI: 10.2140/involve.2017.10.317

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