Orbits of Antichains in Certain Root Posets
Orbits of Antichains in Certain Root Posets
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DOI：
10.37236/7055
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发表时间：
2016-06
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通讯作者：
Chaoping Dong;Suijie Wang
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文献类型：
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作者：
Chaoping Dong;Suijie Wang
本文给出了Propp和Roby定理的另一个证明：偏序集$[m]\times [n]$的任何逆算子轨道上的平均反链长度是$\frac{mn}{m+n}$。可以想象，我们的方法也适用于其他情况。作为一个证明，我们证明了任意逆算符轨道$[m]\times K_{n-1}$中反链的平均尺寸等于$\frac{2 mn}{m+2n-1}$.这里$K_{n-1}$是极小偏序集$[n-1]\oplus（[1] \sqcup [1]）\oplus [n-1]$。注意$[m]\times [n]$和$[m]\times K_{n-1}$可以被解释为某些根偏序集的子族。我们猜想这些根偏序集应该提供一个统一的背景来展示Propp和Roby定义的homomesy现象。
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 root posets should provide a unified setting to exhibit the homomesy phenomenon defined by Propp and Roby.