The degeneration of the Grassmannian into a toric variety and the calculation of the eigenspaces of a torus action

TL;DR

This paper develops a method to compute the dimensions of eigenspaces of a torus action on Grassmannians by degenerating them into toric varieties, providing recursive formulas and verifying results through Euler characteristic calculations.

Contribution

It introduces a recursive approach to determine eigenspace dimensions of torus actions on Grassmannians via degeneration to toric varieties, with validation through Euler characteristic comparisons.

Findings

Recursive formulas for eigenspace dimensions

Verification through Euler characteristic polynomial

Connection between Grassmannian degeneration and toric varieties

Abstract

Using the method of degenerating a Grassmannian into a toric variety, we calculate recursive formulas for the dimensions of the eigenspaces of the action of an n-dimensional torus on a Grassmannian of planes in an n-dimensional space. In order to verify our result we compare it with the polynomial describing the Euler characteristic of invertible sheaves on a projective space with four blown-up points.