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On volume functions of special flow polytopes associated to the root system of type A and On equivariant index of a generalized Bott manifold

Yuki SUGIYAMA 中央大学

2022.12.21

概要

In this thesis, we treat two themes.

In Part I, we consider a flow polytope associated to the root system of type A. The cone spanned by the positive roots is divided into several polyhedral cones called chambers. There is a specific chamber called the nice chamber. In this part, we call a flow polytope for the nice chamber a special flow polytope. Baldoni and Vergne showed the volume function of a special flow polytope is written as an iterated residue. In this case, we show that the volume function satisfies a certain system of differential equations, and conversely, the solution of the system of differential equations is unique up to a constant multiple. In addition, we give an inductive formula for the volume with respect to the rank of the root system of type A.

In Part II, we consider the equivariant index of a generalized Bott manifold. Grossberg and Karshon showed the multiplicity function of the equivariant index for a holomorphic line bundle over a Bott manifold is given by the density function of a twisted cube, which is determined by the structure of the Bott manifold and the line bundle over it. From this, they derived a Demazure-type character formula. In this part, we generalize the above results to generalized Bott manifolds. We show the multiplicity function of the equivariant index is given by the density function of a generalized twisted cube. In addition, we give a Demazure-type character formula of this representation.

参考文献

Part1

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PartⅡ

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[10] Y. Sugiyama, On equivariant index of a generalized Bott manifold, arXiv:2107.12054, 2021. to appear in Osaka J. Math.

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