If $\mathfrak{ g}$ is a Frobenius Lie algebra, then for certain $F\in \mathfrak{ g}^*$ the natural map $\mathfrak{ g}\longrightarrow \mathfrak{ g}^* $ given by $x \longmapsto F[x,-]$ 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.
Categories: math.RA