The Horn cone associated with symplectic eigenvalues

TL;DR

This paper demonstrates that the Horn cone for symplectic eigenvalues shares the same inequalities as the classical Horn cone, with a key inequality replaced by a broader inequality involving traces, revealing a new structural property.

Contribution

It establishes a parallel between the Horn cone for symplectic eigenvalues and the classical Horn cone, highlighting a fundamental inequality difference.

Findings

Horn cone for symplectic eigenvalues has similar inequalities to classical Horn cone

The equality Tr(C) = Tr(A)+Tr(B) is replaced by Tr(C) ≥ Tr(A)+Tr(B)

Provides insight into the structure of symplectic eigenvalues in relation to classical eigenvalues.

Abstract

In this note, we show that the Horn cone associated with symplectic eigenvalues admits the same inequalities as the classical Horn cone, except that the equality corresponding to Tr(C) = Tr(A)+Tr(B) is replaced by the inequality corresponding to Tr(C) $\ge$ Tr(A)+Tr(B).