Mean eigenvalues for simple, simply connected, compact Lie groups

TL;DR

This paper characterizes the regions of mean eigenvalues for certain simple, simply connected compact Lie groups, providing analytical boundary descriptions, area calculations, and insights into the influence of their centers.

Contribution

It offers explicit analytical parameterizations of trace figures for specific Lie groups and computes geometric properties, revealing the role of group centers in eigenvalue distributions.

Findings

Determined regions filled by mean eigenvalues for SU(n), Spin(4n+2), and E6.

Calculated boundary lengths and enclosed areas of trace figures.

Identified cusp points related to the groups' centers and established bounds for eigenvalues of groups with trivial centers.

Abstract

We determine for each of the simple, simply connected, compact and complex Lie groups SU(n), Spin$(4n+2)$ and $E_6$ that particular region inside the unit disk in the complex plane which is filled by their mean eigenvalues. We give analytical parameterizations for the boundary curves of these so-called trace figures. The area enclosed by a trace figure turns out to be a rational multiple of $\pi$ in each case. We calculate also the length of the boundary curve and determine the radius of the largest circle that is contained in a trace figure. The discrete center of the corresponding compact complex Lie group shows up prominently in the form of cusp points of the trace figure placed symmetrically on the unit circle. For the exceptional Lie groups $G_2$, $F_4$ and $E_8$ with trivial center we determine the (negative) lower bound on their mean eigenvalues lying within the real interval $[-1,1]$. We find the rational boundary values -2/7, -3/13 and -1/31 for $G_2$, $F_4$ and $E_8$, respectively.