The unbroken spectrum of type-A Frobenius seaweeds

Abstract

If $\mathfrak{g}$ is a Frobenius Lie algebra, then for certain $F\in {\mathfrak{g}}^{\ast }$ the natural map $\mathfrak{g}⟶{\mathfrak{g}}^{\ast }$ given by $x⟼F[x,-]$ is an isomorphism. The inverse image of $F$ under this isomorphism is called a principal element. We show that if $\mathfrak{g}$ is a Frobenius seaweed subalgebra of $A_{n-1}=\mathfrak{sl}(n)$ then the spectrum of the adjoint of a principal element consists of an unbroken set of integers whose multiplicities have a symmetric distribution. Our proof methods are constructive and combinatorial in nature.