Products of Chern Classes and Chern Numbers on the Permutohedral Variety
Hideya Kuwata

TL;DR
This paper derives explicit formulas for products of Chern classes on the permutohedral variety, enabling precise computation of Chern numbers using combinatorial methods.
Contribution
It provides a new combinatorial approach to express products of Chern classes as multiples of the top Chern class on the permutohedral variety.
Findings
Explicit closed-form formulas for c_k c_{n-k} in terms of c_n.
Calculation of specific Chern numbers for the permutohedral variety.
Demonstration of combinatorial techniques in algebraic geometry.
Abstract
A root system of rank determines an -dimensional smooth projective toric variety associated with the fan of its Weyl chambers. For the root system of type , this variety is the well-known permutohedral variety . Using purely combinatorial methods, we obtain an explicit closed formula expressing the product of Chern classes as a multiple of the top Chern class in the rational cohomology ring . The resulting coefficient, which depends only on and , is given by a closed-form expression. As an application, we compute the Chern number .
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