Toral偏序集与二元谱性质
Toral posets and the binary spectrum property
摘要：
引入了一类偏序集，它生成了$A_{n-1}=\mathfrak{sl}(n)$的李偏序子集代数，其指标可以拓扑实现.特别地，如果$\mathcal{P}$是这样一个\textit{toral poset}，那么它有一个单纯实现，它同伦于$d$单球面的楔形和，其中$d$是相应的A型李偏序集代数$\mathfrak{g}_A(\mathcal{P})$的索引。此外，当$\mathfrak{g}_A(\mathcal{P})$是Frobenius时，它的谱是\textit{binary} ;也就是说，由相等数量的0和1组成。我们还发现，所有对应于最大全序子集基数不超过3的偏序集的Frobenius A型李偏序集代数都有二元谱。
英文摘要：
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.
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