## Involve: A Journal of Mathematics

- Involve
- Volume 10, Number 2 (2017), 317-325.

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

Leonardo C. Mihalcea and Trevor Norton

#### 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 $d=\left({d}_{0},{d}_{1}\right)\in {\mathbb{Z}}_{\ge 0}^{2}$, 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 ${A}_{1}^{1}$.

#### Article information

**Source**

Involve, Volume 10, Number 2 (2017), 317-325.

**Dates**

Received: 13 December 2015

Accepted: 1 April 2016

First available in Project Euclid: 13 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.involve/1513135633

**Digital Object Identifier**

doi:10.2140/involve.2017.10.317

**Mathematical Reviews number (MathSciNet)**

MR3574303

**Zentralblatt MATH identifier**

1350.05180

**Subjects**

Primary: 05E15: Combinatorial aspects of groups and algebras [See also 14Nxx, 22E45, 33C80]

Secondary: 17B67: Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 14M15: Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35]

**Keywords**

affine flag manifolds moment graph curve neighborhood

#### Citation

Mihalcea, Leonardo C.; Norton, Trevor. Combinatorial curve neighborhoods for the affine flag manifold of type $A_1^1$. Involve 10 (2017), no. 2, 317--325. doi:10.2140/involve.2017.10.317. https://projecteuclid.org/euclid.involve/1513135633