论文标题
Coxeter组的平衡常数
Balance constants for Coxeter groups
论文作者
论文摘要
$ 1/3 $ - $ 2/3 $猜想最初于1968年制定，是POSETS理论中最著名的开放问题之一，指出任何非交通订单的余额常数（由线性扩展所确定的数量）至少为$ 1/3 $。通过根据对称组的凸子集重新解释POSET的平衡常数，我们将平衡常数的研究扩展到了任何Coxeter组的凸出子集$ c $。值得注意的是，我们猜想$ 1/3 $的下限仍然适用于任何有限的Weyl组，并出现了新的和有趣的平等案例。 我们将几个主要结果概括为$ 1/3 $ - $ 2/3 $猜想到这个新环境：当$ c $是一个低于任何无循环coxeter群体中完全交换元素的弱订单间隔时，我们证明了我们的猜想（我们在宽度posets的案例中的概括）（我们对所有有限级订单的均匀订单）进行了统一的订单，以供应均匀的订单，以使所有有限级的订单中的均匀订单供应均匀的订单，以下是均匀的订单。我们解决猜想的广义半月。 我们希望这种新的观点可以阐明考虑$ 1/3 $ - $ 2/3 $猜想的适当一般性，因此在哪种方法上可能会成功解决它。
The $1/3$-$2/3$ Conjecture, originally formulated in 1968, is one of the best-known open problems in the theory of posets, stating that the balance constant (a quantity determined by the linear extensions) of any non-total order is at least $1/3$. By reinterpreting balance constants of posets in terms of convex subsets of the symmetric group, we extend the study of balance constants to convex subsets $C$ of any Coxeter group. Remarkably, we conjecture that the lower bound of $1/3$ still applies in any finite Weyl group, with new and interesting equality cases appearing. We generalize several of the main results towards the $1/3$-$2/3$ Conjecture to this new setting: we prove our conjecture when $C$ is a weak order interval below a fully commutative element in any acyclic Coxeter group (an generalization of the case of width-two posets), we give a uniform lower bound for balance constants in all finite Weyl groups using a new generalization of order polytopes to this context, and we introduce generalized semiorders for which we resolve the conjecture. We hope this new perspective may shed light on the proper level of generality in which to consider the $1/3$-$2/3$ Conjecture, and therefore on which methods are likely to be successful in resolving it.
