Diagram monoids and Graham-Houghton graphs: Idempotents and generating sets of ideals
Diagram monoids and Graham-Houghton graphs: Idempotents and generating sets of ideals
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DOI：
10.1016/j.jcta.2016.09.001
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发表时间：
2014-04
期刊：
影响因子：
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通讯作者：
J. East;R. Gray
中科院分区：
文献类型：
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作者：
J. East;R. Gray
我们研究了分拆、Brauer和Jones么半群的理想，通过分析它们的Graham-Houghton图，建立了关于生成集和幂等生成集的各种组合结果。证明了划分么半群Pn的每个真理想都是幂等元生成半群，并得到了生成这些半群所需的最小元素个数(和最小幂等元个数)的公式。特别地，我们证明了这两个分别称为半群的秩和幂等数的数是相等的，并刻画了这个最小基数的生成集。刻画和计数了Pn的最大真理想的极小幂等元生成集，它与Pn的奇异部分重合.对Brauer么半群和Jones么半群的理想也证明了类似的结果；在这两种情况下，秩和幂等元的秩变得相等，并刻画了所有的最小生成集.我们还证明了当应用于相应的扭曲半群代数(分拆、Brauer和Temperley-Lieb代数)时，所得到的秩和幂等秩结果如何允许人们恢复它们的胞模(被视为胞代数)的维度的公式，在半简单的情况下，这些公式是关于代数的不可约表示的维度的公式。除了具有代数意义外，我们的结果还涉及图论中几个研究得很好的主题，包括完全匹配的计数问题(与计算{0，1}-矩阵的恒等式和Pfaffian定向理论有关)，以及求Johnson图的因子分解的问题。我们的结果还结合了几个著名的数字序列，如Stirling，Bell，Catalan和Fibonacci数。
We study the ideals of the partition, Brauer, and Jones monoid, establishing various combinatorial results on generating sets and idempotent generating sets via an analysis of their Graham–Houghton graphs. We show that each proper ideal of the partition monoid P n is an idempotent generated semigroup, and obtain a formula for the minimal number of elements (and the minimal number of idempotent elements) needed to generate these semigroups. In particular, we show that these two numbers, which are called the rank and idempotent rank (respectively) of the semigroup, are equal to each other, and we characterize the generating sets of this minimal cardinality. We also characterize and enumerate the minimal idempotent generating sets for the largest proper ideal of P n, which coincides with the singular part of P n. Analogous results are proved for the ideals of the Brauer and Jones monoids; in each case, the rank and idempotent rank turn out to be equal, and all the minimal generating sets are described. We also show how the rank and idempotent rank results obtained, when applied to the corresponding twisted semigroup algebras (the partition, Brauer, and Temperley–Lieb algebras), allow one to recover formulae for the dimensions of their cell modules (viewed as cellular algebras) which, in the semisimple case, are formulae for the dimensions of the irreducible representations of the algebras. As well as being of algebraic interest, our results relate to several well-studied topics in graph theory including the problem of counting perfect matchings (which relates to the problem of computing permanents of {0, 1}-matrices and the theory of Pfaffian orientations), and the problem of finding factorizations of Johnson graphs. Our results also bring together several well-known number sequences such as Stirling, Bell, Catalan and Fibonacci numbers.