Osaka Journal of Mathematics

Weyl group symmetry on the GKM graph of a GKM manifold with an extended Lie group action

Shizuo Kaji

Full-text: Open access


We consider the class of manifolds with compact Lie group actions which restrict to GKM-actions on the maximal torus. First, we see their GKM-graphs admit symmetry of the Weyl groups. And then, we study its combinatorial abstraction; starting with abstract GKM-graphs with symmetry, we derive certain properties which reflect topology in a purely combinatorial way.

Article information

Osaka J. Math., Volume 52, Number 1 (2015), 31-43.

First available in Project Euclid: 24 March 2015

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55N91: Equivariant homology and cohomology [See also 19L47]
Secondary: 57S15: Compact Lie groups of differentiable transformations


Kaji, Shizuo. Weyl group symmetry on the GKM graph of a GKM manifold with an extended Lie group action. Osaka J. Math. 52 (2015), no. 1, 31--43.

Export citation


  • J.C. Becker and D.H. Gottlieb: The transfer map and fiber bundles, Topology 14 (1975), 1–12.
  • G. Brumfiel and I. Madsen: Evaluation of the transfer and the universal surgery classes, Invent. Math. 32 (1976), 133–169.
  • M. Demazure: Invariants symétriques entiers des groupes de Weyl et torsion, Invent. Math. 21 (1973), 287–301.
  • M. Goresky, R. Kottwitz and R. MacPherson: Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math. 131 (1998), 25–83.
  • V. Guillemin and C. Zara: 1-skeleta, Betti numbers, and equivariant cohomology, Duke Math. J. 107 (2001), 283–349.
  • S. Kaji: Schubert calculus, seen from torus equivariant topology, Trends in Mathematics – New Series 12, (2010), 71–90.
  • S. Kaji: Equivariant Schubert calculus of Coxeter groups, Proc. Steklov Inst. Math., 275 (2011), 239–250
  • B. Kostant and S. Kumar: The nil Hecke ring and cohomology of $G/P$ for a Kac–Moody group $G$, Adv. in Math. 62 (1986), 187–237.
  • S. Kuroki: GKM graphs induced by GKM manifolds with $\mathit{SU}(l+1)$-symmetries, Trends in Mathematics – New Series 12, (2010), 103–113.
  • M. Masuda: Symmetry of a symplectic toric manifold, J. Symplectic Geom. 8 (2010), 359–380.
  • M. Masuda and T. Panov: On the cohomology of torus manifolds, Osaka J. Math. 43 (2006), 711–746.
  • M. Wiemeler: Torus manifolds with non-abelian symmetries, Trans. Amer. Math. Soc. 364 (2012), 1427–1487.