根偏序集及其富反链

The root posets and their rich antichains

令?为一个秩为n(n≥2)的(连通)Dynkin图(diagram),令Φ_+=Φ_+(?)为对应的根偏序集(root poset)(它由对应于一个固定的根基(root basis)的所有正根构成).Φ_+的宽度(width)是n.本文将证明Φ_+是"圆锥形的"(conical),即它是n个实心链(solid chain)的非交并.Φ_+中的富反链(rich antichain)是基数(cardinality)为n-1的反链.众所周知,富反链的个数等于Φ_+的基数.Φ_+中富反链的集合R(?)自身可看作是一个偏序集,它与Φ_+相似,却并不总是同构于Φ_+.本文将证明,总是存在唯一的富反链A使得任意富反链都包含在由A生成的理想中.对于?≠E_6,A中所有根的长度都相同,即为e_2,其中e_1≤e_2≤···≤e_n是?的指数(exponent).对于?=E_6,反链A包含四个长度为e_2=4的根和一个长度为5的根.

Let △be a（connected）Dynkin diagram of rank n≥2 and Φ＋=Φ＋（△）the corresponding root poset（it consists of all positive roots with respect to a fixed root basis）.The width of Φ＋is n.We will show thatΦ＋is＂conical＂：It is the disjoint union of n solid chains.The rich antichains inΦ＋are the antichains of cardinality n-1.It is well known that the number of rich antichains is equal to the cardinality of Φ＋.The set R（△）of rich antichains inΦ＋can itself be considered as a poset which is quite similar,but not always isomorphic,toΦ＋.We will show that there always exists a unique rich antichain A such that any rich antichain is contained in the ideal generated by A.For△≠E6 all roots in A have the same length,namely e2,where e1≤e2≤···≤en are the exponents of△.For△=E6,the antichain A consists of four roots of length e2=4 and one root of length 5.