Toral posets and the binary spectrum property

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 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 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.