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

TL;DR

This paper explicitly determines the cohomology ring structure of GKM graphs associated with flag manifolds of classical types, providing a clearer algebraic understanding of their topological and geometric properties.

Contribution

It offers an elementary method to compute the ring structure of GKM graph cohomology for classical flag manifolds with integer coefficients.

Findings

Ring structure explicitly determined for type A, B, D flag manifolds.

Ring structure with 2-inverted coefficients determined for type C.

Provides algebraic tools for understanding equivariant cohomology of flag manifolds.

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 $H^*_T(M)$ 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.