论文标题
关于无向Bruhat图的自身形态学
On automorphisms of undirected Bruhat graphs
论文作者
论文摘要
（定向的）bruhat图$ \hatγ（u，v）$具有bruhat间隔$ [u，v] $作为顶点的元素，并带有反射乘以乘法给定的边缘。著名的是，$ \hatγ（e，v）$是常规的，并且仅当舒伯特品种$ x_v $平滑时，$ v $上的这种条件的特征是避免图案。在这项工作中，我们对无方向的bruhat图$γ（e，v）$进行分类；出乎意料的是，这类排列也以避免模式为特征，并且在平滑排列和自偶置排列的类别之间很好地位于。 This leads us to a general investigation of automorphisms of $Γ(u,v)$ in the course of which we show that special matchings, which originally appeared in the theory Kazhdan--Lusztig polynomials, can be characterized as certain $Γ(u,v)$-automorphisms which are conjecturally sufficient to generate the orbit of $e$ under $Aut(Γ(e,v))$.
The (directed) Bruhat graph $\hatΓ(u,v)$ has the elements of the Bruhat interval $[u,v]$ as vertices, with directed edges given by multiplication by a reflection. Famously, $\hatΓ(e,v)$ is regular if and only if the Schubert variety $X_v$ is smooth, and this condition on $v$ is characterized by pattern avoidance. In this work, we classify when the undirected Bruhat graph $Γ(e,v)$ is vertex-transitive; surprisingly this class of permutations is also characterized by pattern avoidance and sits nicely between the classes of smooth permutations and self-dual permutations. This leads us to a general investigation of automorphisms of $Γ(u,v)$ in the course of which we show that special matchings, which originally appeared in the theory Kazhdan--Lusztig polynomials, can be characterized as certain $Γ(u,v)$-automorphisms which are conjecturally sufficient to generate the orbit of $e$ under $Aut(Γ(e,v))$.
