Compactification of the isospectral varieties of nilpotent Toda lattices

TL;DR

This paper introduces a method to compactify isospectral varieties of nilpotent Toda lattices using unipotent group orbits, analyzes their topology through cell decompositions, and finds similarities with Schubert varieties.

Contribution

It constructs a chain complex for these varieties, computes their rational cohomology, and reveals patterns in Betti numbers, linking them to Schubert varieties.

Findings

Rational cohomology patterns are classified into three types.

Compactified varieties resemble smooth Schubert varieties in cell structure.

For type A Lie algebras, cohomology matches that of Schubert varieties.

Abstract

The paper concerns a compactification of the isospectral varieties of nilpotent Toda lattices for real split simple Lie algebras. The compactification is obtained by taking the closure of unipotent group orbits in the flag manifolds. The unipotent group orbits are called the Peterson varieties and can be used in the complex case to describe the quantum cohomology of Grassmannian manifolds. We construct a chain complex based on a cell decomposition consisting of the subsystems of Toda lattices. Explicit formulae for the incidence numbers of the chain complex are found, and encoded in a graph containing an edge whenever an incidence number is non-zero. We then compute rational cohomology, and show that there are just three different patterns in the calculation of Betti numbers. Although these compactified varieties are singular, they resemble certain smooth Schubert varieties e.g. they both have a cell decomposition consiting of unipotent group orbits of the same dimensions. In particular, for the case of a Lie algebra of type $A$ the rational homology/cohomology obtained from the compactified isospectral variety of the nilpotent Toda lattice equals that of the corresponding Schubert variety.