Root systems of torus graphs

Torus manifold is a compact oriented $2n$-dim $T^n$-manifold with fixed points. From torus manifold, we can define a labelled graph as follows:
– vertices are fixed points;
– edges are one dimensional orbits;
– edges are labelled by tangential representations around fixed points.
This labelled graph is called a torus graph.
It is known that we can compute the equivariant cohomology of torus manifold by using combinatorial structure of torus graph. In this talk, we define root systems on torus graph and characterize what kind of compact connected non-abelian Lie group (whose maximal torus is $T^n$) acts on torus manifold.
This is a joint work with M. Masuda.