Homogeneous Einstein metrics on the generalized flag manifold Sp(n)/(U(p) x U(n-p))

TL;DR

This paper determines the exact count of non-Kähler $Sp(n)$-invariant Einstein metrics on a specific class of generalized flag manifolds, using polynomial system analysis.

Contribution

It provides a precise enumeration of Einstein metrics on $Sp(n)/(U(p) imes U(n-p))$, advancing understanding of invariant Einstein metrics on flag manifolds.

Findings

Exact number of non-Kähler Einstein metrics identified

Analysis of polynomial systems applied to geometric classification

Insights into the structure of invariant Einstein metrics

Abstract

We find the precise number of non-K\"ahler $Sp(n)$-invariant Einstein metrics on the generalized flag manifold $M=Sp(n)/(U(p)\times U(n-p))$ with $n\geq 3$ and $1\leq p\leq n-1$. We use an analysis on parametric systems of polynomial equations and we give some insight towards the study of such systems.