The Betti numbers of real toric varieties associated to Weyl chambers of types \(E_7\) and \(E_8\) (Q6568748)

Toric topology plays an important role in a wide variety of mathematical contexts and has been a hot topic for the past twenty-five years. A core object in Toric topology has been the moment angle complex, which has been studied by Buchstaber, Panov, etc. Later, the theory of moment angle complexes was applied to toric varieties. As a result, toric varieties have also become involved in toric topology.\N\NIn general, the methods of research in toric topology can be summarized as follows:\N\N1. Algebraically: This type of papers concentrates on using algebraic computations to obtain a solution. A classical way is to compute spectral sequences. \N\N2. Combinatorially: This type of papers is more likely to verify the homotopy type for some special spaces or complexes, or, more generally, using some properties to verify a combinatorial object. Such papers mainly involve computations on simplicial complexes, the characteristic matrix, etc.\N\N3. Considering Manifolds: Because many objects in toric topology are manifolds, some ideas from manifold theory are also useful. Such research frequently can be seen in equivariant cohomology, the toral rank conjecture, configuration spaces etc.\N\NThis paper is to complete the computation of the Betti numbers of the real locus complete fan generated by a root system of a Lie group. The fundamental result is a Hochster-type formula (Theorem 2.1. in the paper). This formula means it's enough to consider the full subcomplex generated by Weyl chambers. By observing that such a complex is transitive and by using the MV sequence some vertices can be removed, this problem can be converted to a problem which can be solved by computer using the algorithm in Theorem 3.1. The authors also give their code in github, which is convenient.