Supersolvability and the Koszul property of root ideal arrangements
Supersolvability and the Koszul property of root ideal arrangements
复制标题
根理想排列的超解性和 Koszul 性质
DOI：
10.1090/proc/12810
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发表时间：
2014
影响因子：
0.9
通讯作者：
Axel Hultman
中科院分区：
文献类型：
--
作者：
Axel Hultman
根理想排列A_I是对应于有限晶体根系的正根上的根偏序集的理想阶I中的根的反射超平面的集合。获得了超解根理想排列的表征。也就是说，当且仅当 I 是链可剥离时，A_I 是超可解的，这意味着可以通过在每个步骤中删除也是阶过滤器的最大链来从 I 到达空偏序集。特别是，在采用子理想的情况下，超可解性得以保留。我们确定与不可超解排列相对应的最大理想。本质上有两种这样的理想，一种是 D_4 类型，另一种是 F_4 类型。通过证明如果 I 包含其中之一，则 A_I 不是线闭合的，我们推断 Orlik-Solomon 代数 OS(A_I) 具有 Koszul 性质当且仅当 A_I 是超可解的。
A root ideal arrangement A_I is the set of reflecting hyperplanes corresponding to the roots in an order ideal I of the root poset on the positive roots of a finite crystallographic root system. A characterisation of supersolvable root ideal arrangements is obtained. Namely, A_I is supersolvable if and only if I is chain peelable, meaning that it is possible to reach the empty poset from I by in each step removing a maximal chain which is also an order filter. In particular, supersolvability is preserved under taking subideals. We identify the maximal ideals that correspond to non-supersolvable arrangements. There are essentially two such ideals, one in type D_4 and one in type F_4. By showing that A_I is not line-closed if I contains one of these, we deduce that the Orlik-Solomon algebra OS(A_I) has the Koszul property if and only if A_I is supersolvable.