The cohomology ring of the GKM graph of a flag manifold of classical type

Abstract

If a closed smooth manifold $M$ with an action of a torus $T$ satisfies certain conditions, then a labeled graph $\mG_M$ with labeling in $H^2(BT)$ is associated with $M$, which encodes a lot of geometrical information on $M$. For instance, the "graph cohomology" ring $\mHT^*(\mG_M)$ of $\mG_M$ is defined to be a subring of $\bigoplus_{v\in V(\mG_M)}H^*(BT)$, where $V(\mG_M)$ is the set of vertices of $\mG_M$, and is known to be often isomorphic to the equivariant cohomology $H_T^{\ast }(M)$ of $M$. In this paper, we determine the ring structure of $\mHT^*(\mG_M)$ with $\mathbb{Z}$ (resp. $\mathbb{Z}[1/2]$) coefficients when $M$ is a flag manifold of type A, B or D (resp. C) in an elementary way.

22 pages

References in corpus (1)

• The graph cohomology ring of the GKM graph of a flag manifold of type $G_2$

Cited by in corpus (7)

• The equivariant cohomology rings of Peterson varieties in all Lie types
• The equivariant cohomology rings of Peterson varieties
• Upper bounds for the dimension of tori acting on GKM manifolds
• The torus equivariant cohomology rings of Springer varieties
• The $T$-equivariant integral cohomology ring of $F_4/T$
• GKM graph locally modeled by $T^n\times S^1$-action on $T^{\ast }{\mathbb{C}}^n$ and its graph equivariant cohomology
• Equivariant $K$-theory of even-dimensional complex quadrics