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Axial functions on complete graphs
Fix a graph Γ. A naive question is how many axial functions exist on Γ. It seems that this question has not been considered so far. In this talk, we make a small step toward this general question. Indeed, we consider the question when Γ is a complete graph Kn with n nodes. We show that an axial function on Kn is standard in some sense when the rank m of the acting torus is larger than [n/2], but this is not the case when m = [n/2] and n is even. This is joint work with Shintaro Kuroki and Hiroshi Maeda.
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Integral cohomology ring of 3-dimensional Gelfand–Zetlin toric variety
We compute the integral cohomology ring of the toric variety X associated with the three dimensional Gelfand–Zetlin polytope GZ(3). This variety is singular that is not an orbifold. Nevertheless, its odd-degree cohomology vanishes, which identifies its equivariant cohomology with the ring of piecewise polynomials on GZ(3). Using this description, we exhibit an additive basis of the integral cohomology of X and determine its multiplicative structure completely. This work is an explicit case study of several general questions.