Antichains of positive roots and Heaviside functions

Orbits of antichains in certain root posets

This paper gives another proof of Propp and Roby's theorem saying that the average antichain size in any reverse operator orbit of the poset $[m]\times [n]$ is $\frac{mn}{m+n}$. It is conceivable that our method should work for other situations. As a demonstration, we show that the average size of antichains in any reverse operator orbit of $[m]\times K_{n-1}$ equals $\frac{2mn}{m+2n-1}$. Here $K_{n-1}$ is the minuscule poset $[n-1]\oplus ([1] \sqcup [1]) \oplus [n-1]$. Note that $[m]\times [n]$ and $[m]\times K_{n-1}$ can be interpreted as sub-families of certain root posets. We guess these r…

Short antichains in root systems, semi-Catalan arrangements, and B-stable subspaces

Let $\be$ be a Borel subalgebra of a complex simple Lie algebra $\g$. An ideal of $\be$ is called ad-nilpotent, if it is contained in $[\be,\be]$. The generators of an ad-nilpotent ideal give rise to an antichain in the poset of positive roots, and the whole theory can be expressed in a combinatorial fashion, in terms of antichains. The aim of this paper is to present a refinement of the enumerative theory of ad-nilpotent ideals for the case in which $\g$ has roots of different length. An antichain is called short, if it consists of short roots. We obtain, for short antichains, analogues of al…

On orbits of antichains of positive roots

For any finite poset P, there is a natural operator $X$ acting on the antichains of P. We discuss conjectural properties of this operator for some graded posets associated with irreducible root systems. In particular, if $Δ^+$ is the set of positive roots and $Π$ is the set of simple roots in $Δ^+$, then we consider the cases $P=Δ^+$ and $Δ^+\setminus Π$. For the root system of type $A_n$, we consider an $X$-invariant integer-valued function on the set of antichains of $Δ^+$ and establish some properties of it.

The root posets and their rich antichains

Let $Δ$ be a (connected) Dynkin diagram of rank $n\ge 2$ and $Φ_+ = Φ_+(Δ)$ 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 that $Φ_+$ is "conical": it is the disjoint union of $n$ solid chains. The rich antichains in $Φ_+$ 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 $\mathcal R(Δ)$ of rich antichains in $Φ_+$ can itself be considered as a poset which is quite similar, but not always isomorphic, to $Φ_+$. We …

Narayana数的对称性与根偏序的行空化

Symmetry of Narayana numbers and rowvacuation of root posets

本文证明对于经典类型的根偏序，行空化算子实现了Panyushev提出的对合，从而展示了W-Narayana数的对称性。

关于若干无限偏序集族中理想与反链的平均大小

On the Average Sizes of Ideals and Antichains in Some Infinite Families of Posets

研究有根无标号树生成的偏序集族中理想与反链的平均大小，利用生成函数和奇点分析证明二次根分解族中理想平均大小为 n/3，并分析反链常数及分支限制的影响。