## Involve: A Journal of Mathematics

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

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

#### Abstract

Let $X$ be the affine flag manifold of Lie type $A11$. Its moment graph encodes the torus fixed points (which are elements of the infinite dihedral group $D∞$) and the torus stable curves in $X$. Given a fixed point $u ∈ D∞$ and a degree $d = ( d 0 , d 1 ) ∈ ℤ ≥ 0 2$, 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 $≤ d$. 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

#### 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

#### References

• A. S. Buch and L. C. Mihalcea, “Curve neighborhoods of Schubert varieties”, J. Differential Geom. 99:2 (2015), 255–283.
• V. G. Kac, Infinite-dimensional Lie algebras, 2nd ed., Cambridge University Press, 1985.
• S. Kumar, Kac–Moody groups, their flag varieties and representation theory, Progress in Mathematics 204, Birkhäuser, Boston, 2002.
• L. Mare and L. C. Mihalcea, “An affine quantum cohomology ring for flag manifolds and the periodic Toda lattice”, preprint, 2014.