Representations of $\mathfrak{sl}( 2,\mathbb{C} )$ on Posets and the Sperner Property
Representations of $\mathfrak{sl}( 2,\mathbb{C} )$ on Posets and the Sperner Property
复制标题
$mathfrak{sl}( 2,mathbb{C} )$ 在偏序集和 Sperner 性质上的表示
DOI：
--
复制
发表时间：
1982
期刊：
影响因子：
--
通讯作者：
Robert A. Proctor
中科院分区：
文献类型：
--
作者：
Robert A. Proctor
一个排序偏序集被称为Sperner，如果它没有比它的最大秩更大的反链。给出了秩偏序集是秩对称的、秩单峰的和强Sperner的一个充要条件。这个条件涉及$\mathfrak{sl}（2，\mathbb {C}）$的表示。它用于提供一个新的、简短的证明，证明这种性质的组合在乘积运算下得到了保留。这个条件的充分部分也被用来提供新的，更简单的证明，某些combinatorially有趣的偏序集是秩对称，秩单峰和强Sperner。
A ranked partially ordered set is said to be Sperner if it has no antichain bigger than its largest rank. A necessary and sufficient condition for a ranked partially ordered set to be rank symmetric, rank unimodal and strongly Sperner is presented. This condition involves representations of $\mathfrak{sl} ( 2,\mathbb{C} )$. It is used to provide a new, short proof that this combination of properties is preserved under the product operation. The sufficient part of this condition is also used to provide new, simpler proofs that certain combinatorially interesting partially ordered sets are rank symmetric, rank unimodal and strongly Sperner.