Combinatorics of minuscule representations (Cambridge Tracts in Mathematics 199) By R. M. Green
Combinatorics of minuscule representations (Cambridge Tracts in Mathematics 199) By R. M. Green
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微小表示的组合（剑桥数学丛书 199） 作者：R. M. Green
DOI：
10.1112/blms/bdu112
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发表时间：
2015
影响因子：
0.9
通讯作者：
V. Reiner
中科院分区：
文献类型：
--
作者：
V. Reiner
是复单李代数的极小表示||是“尽可能小”：不仅是不可约的和有限维的，而且在Weyl群的作用下，所有的权重都在同一个轨道上|| .特别是，最高权重||有一维权空间，所以它的所有权空间都是一维的。结果表明，对权重集进行部分排序是有效的||，定义权偏序集，其中||当||是正根的非负和。正如RM绿色的书中所解释的，偏序集结构是极小表示的许多特殊性质的关键，使其比典型的表示更简单。||- 不可约。极小表示只存在于类型中|| .型||他们是外部势力||自然的表象||对于李代数||零矩阵的迹|| .这里的外尔群||型||可以看作是对称群||置换标准基|| .单项楔||给出一个权向量的基础，||，在此描述为||，按照权重偏序集排序：
To be a minuscule representation of a complex simple Lie algebra|| is to be ‘as small as possible’: not only irreducible and finite-dimensional, but with all weights in the same orbit under the action of the Weyl group||. In particular, the highest weight|| has one-dimensional weight space, so all of its weight spaces are one-dimensional. It turns out to be fruitful to partially order the set of weights||, defining the weight poset in which|| when|| is a non-negative sum of positive roots. As explained in RM Green's book under review here, poset structures are the key to a minuscule representation's many special properties, making it simpler than a typical||-irreducible.Minuscule representations exist only in types||. In type|| they are the exterior powers|| of the natural representation|| for the Lie algebra|| of trace zero matrices in||. Here the Weyl group|| of type|| can be taken to be the symmetric group|| permuting the standard basis||. The monomial wedges|| give a basis of weight vectors for||, depicted here for||, ordered as in the weight poset: