The \(T\)-equivariant integral cohomology ring of \(F_4/T\) (Q482144)

Let \(G\) be a simply connected semi-simple compact Lie group and let \(T\) be its maximal torus. The homogeneous space \(G/T\) is a complex projective flag variety and we know that it plays an important role in topology, algebraic geometry, representation theory, intersection theory and combinatorics. In particular, the \(T\)-equivariant integral cohomology ring \({H^{*}}_T (G/T) = H^{*} (ET \times_T G/T)\) is especially important, where \(ET\) is the universal space of the universal principal bundle \(T\rightarrow ET \rightarrow ET/T = BT\) with bundle group \(T\), and \(T\) acts on \(G/T\) by the left multiplication. \textit{M. Goresky} et al. [Invent. Math. 131, No. 1, 25--83 (1998; Zbl 0897.22009)] gave a powerful method to determine the equivariant rational cohomology of some good spaces. It is called the GKM theory. Let us explain what it is. Since \(T\) acts on \(G/T\) by the left multiplication, then the fixed point set \((G/T)^T\) is identified with the Weyl group \(W\) which is the symmetry group of the root system of \(G\), hence the inclusion \(i:(G/T)^T \rightarrow G/T\) induces the following contravariant morphism \[ i^* : {H^{*}}_T (G/T) \rightarrow {H^{*}}_T ((G/T)^T) = \prod_{w\in W} H^{*}(BT) = Map(W, H^*(BT)), \] where \(BT\) is the classifying space of \(T\). Upon tensoring with rationals \(\mathbf Q\), \(i^*\) is injective by the localization theorem. The GKM theory gives a way to describe the image of this map \(i^*\), which was restated by \textit{V. Guillemin} and \textit{C. Zara} [Duke Math. J. 107, No. 2, 283--349 (2001; Zbl 1020.57013)] as follows. The image of \(i^*\) is completely determined by a graph with additional data obtained from \(G\). Precisely they defined the cohomology ring of the graph as a subring of \((Map(W, H^*(BT)))\) and showed that it coincides with the image of \(i^*\). This graph is called a GKM graph. \textit{M. Harada} et al. [Adv. Math. 197, No. 1, 198--221 (2005; Zbl 1110.55003)] showed that, with integer coefficients, \(i^*\) is injective and its image coincides with the cohomology of the GKM graph. By concrete computations by the GKM theory, for a simple Lie group \(G\) of classical types and of type \(G_2\), \textit{Y. Fukukawa, H. Ishida} and \textit{M. Masuda} [``The cohomology ring of the GKM graph of a flag manifold of classical type'', preprint, \url{arXiv:1104.1832v3} [math.AT]] and \textit{Y. Fukukawa} [``The graph cohomology ring of the GKM graph of a flag manifold of type G2'', preprint, \url{arXiv:1207.5229v1} [math.AT]] determined the cohomology ring of the GKM graph of \(G/T\). Hence they determined the equivariant integral cohomology ring \({H^*}_T (G/T)\) for a Lie group \(G\) of types \(A, B, D\), and \(G_2\). In this paper the author explicitly calculate the \(T\)-equivariant (and also ordinary) integral cohomology ring of the exceptional Lie group \(F_4 /T\) by the GKM theory, symmetric functions and Serre spectral sequence.