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

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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 ) 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

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


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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.