Toral posets and the binary spectrum property

TL;DR

This paper introduces toral posets that generate Lie subalgebras with topologically realizable index and shows that Frobenius cases have binary spectra, especially for posets with small maximal chains.

Contribution

It defines toral posets and links their topological properties to the spectral characteristics of associated Lie algebras, revealing new structural insights.

Findings

Toral posets have simplicial realizations homotopic to wedge sums of spheres.

Frobenius Lie algebras associated with these posets have binary spectra.

Posets with maximal chains of size at most three produce Frobenius algebras with binary spectra.

Abstract

We introduce a family of posets which generate Lie poset subalgebras of $A_{n-1}=\mathfrak{sl}(n)$ whose index can be realized topologically. In particular, if $\mathcal{P}$ is such a \textit{toral poset}, then it has a simplicial realization which is homotopic to a wedge sum of $d$ one-spheres, where $d$ is the index of the corresponding type-A Lie poset algebra $\mathfrak{g}_A(\mathcal{P})$. Moreover, when $\mathfrak{g}_A(\mathcal{P})$ is Frobenius, its spectrum is \textit{binary}; that is, consists of an equal number of 0's and 1's. We also find that all Frobenius, type-A Lie poset algebras corresponding to a poset whose largest totally ordered subset is of cardinality at most three have a binary spectrum.