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

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 $H^*_T(M)$ of $M$. In this paper, we determine the ring structure of \mHT^*(\mG_M) with $\Z$ (resp. $\Z[1/2]$) coefficients when $M$ is a flag manifold of type A, B or D (resp. C) in an elementary way.Comment: 22 page