格格 congruences、扇形和 Hopf 代数
Lattice congruences, fans and Hopf algebras
AI总结：
本文基于排列到三角剖分和子集的映射，统一解释其几何与代数性质，并推广至Coxeter群的格congruence，构建具有强性质的凸锥扇形，揭示子Hopf代数的基结构，以及通过无限序列极限构建Malvenuto-Reutenauer代数。
摘要：
我们给出两个著名映射的几何和代数性质的统一解释，一个是从排列到三角剖分的映射，另一个是从排列到子集的映射。此外，我们对这些映射进行了广泛的推广。具体而言，对于任何弱顺序上 Coxeter 组的格 congruence，我们构造了一个具有强性质的凸锥完整扇形。我们证明，如果一个在对称群上的格 congruence 家族满足某些兼容条件，则该家族定义了 permutation 的 Malvenuto-Reutenauer Hopf 代数的子代数。此类子代数具有由某种模式避免描述的基。应用这些结果，我们构建 Malvenuto-Reutenauer 代数作为无限序列中较小代数的极限，其中第二个代数是非交换对称函数的 Hopf 代数。我们还将一个出现似乎与 Baxter 排列等数的排列集与一个扇形和 Hopf 代数相关联。
英文摘要：
We give a unified explanation of the geometric and algebraic properties of two well-known maps, one from permutations to triangulations, and another from permutations to subsets. Furthermore we give a broad generalization of the maps. Specifically, for any lattice congruence of the weak order on a Coxeter group we construct a complete fan of convex cones with strong properties relative to the corresponding lattice quotient of the weak order. We show that if a family of lattice congruences on the symmetric groups satisfies certain compatibility conditions then the family defines a sub Hopf algebra of the Malvenuto-Reutenauer Hopf algebra of permutations. Such a sub Hopf algebra has a basis which is described by a type of pattern avoidance. Applying these results, we build the Malvenuto-Reutenauer algebra as the limit of an infinite sequence of smaller algebras, where the second algebra in the sequence is the Hopf algebra of non-commutative symmetric functions. We also associate both a fan and a Hopf algebra to a set of permutations which appears to be equinumerous with the Baxter permutations.
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