The unbroken spectrum of type-A Frobenius seaweeds

TL;DR

This paper investigates the spectrum of principal elements in Frobenius seaweed subalgebras of a1a1, revealing an unbroken integer spectrum with symmetric multiplicities, using combinatorial and constructive methods.

Contribution

It demonstrates that the spectrum of principal elements in Frobenius seaweed subalgebras of a1a1 is an unbroken set of integers with symmetric multiplicities, providing a new combinatorial approach.

Findings

Spectrum of principal elements is an unbroken set of integers.

Multiplicity of eigenvalues is symmetrically distributed.

Constructive and combinatorial proof methods are used.

Abstract

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.