论文标题
pop-tsack扭转的动力
Dynamics of Pop-Tsack Torsing
论文作者
论文摘要
对于有限的不可约Coxeter组$（W，s）$，带有固定的Coxeter元素$ c $和一组反射$ T $，Defant和Williams定义了Pop-Tsack Torsing操作$ \ Mathrm {popt} \ colon w \ w $ w $ w $由$ \ mathrm {popt}（popt}（w） $π_t（w）= \ bigVee_ {t \ leq_ {t} W，\ t} in t}^{nc（w，c）} t $是所有反射的联接，所有反射的联合是在非交叉的lattice lattice lattice $ nc（w，c）中以绝对顺序以$ w $下的$ w $下列。这是Defant引入的POP堆栈排序运算符$ \ Mathrm {pops} $的“双重”概念，是一种将POP堆栈分类运算符在$ \ mathfrak {s} _n $上概括为通用Coxeter组的方式。将元素$ w \在w $中定义为$ o _ {\ mathrm {popt}}}（w）= \ {w，\ mathrm {popt}（w），\ mathrm {popt}^2（w）^2（w），\ ldots \ \} $。 Defant和Williams确定了最长可能的前向轨道的长度$ \ max_ {w \ in W} | O _ {\ Mathrm {popt}}}（w）| $ coxeter类型的Coxeter组和类型$ d $ ty -type $ d $。在他们的论文中，他们还提出了多个关于枚举轨道长度接近的元件的猜想。我们解决了他们对枚举提出的所有猜想，在此过程中，我们将$ a，b $和$ d $的这些元素的这些要素完全分类，并具有接近最大的轨道长度。
For a finite irreducible Coxeter group $(W,S)$ with a fixed Coxeter element $c$ and set of reflections $T$, Defant and Williams define a pop-tsack torsing operation $\mathrm{Popt}\colon W \to W$ given by $\mathrm{Popt}(w) = w \cdot π_T(w)^{-1}$ where $π_T(w) = \bigvee_{t \leq_{T}w, \ t \in T}^{NC(w,c)}t$ is the join of all reflections lying below $w$ in the absolute order in the non-crossing partition lattice $NC(w,c)$. This is a "dual" notion of the pop-stack sorting operator $\mathrm{Pops}$ introduced by Defant as a way to generalize the pop-stack sorting operator on $\mathfrak{S}_n$ to general Coxeter groups. Define the forward orbit of an element $w \in W$ to be $O_{\mathrm{Popt}}(w) = \{w, \mathrm{Popt}(w), \mathrm{Popt}^2(w), \ldots \}$. Defant and Williams established the length of the longest possible forward orbits $\max_{w \in W}|O_{\mathrm{Popt}}(w)|$ for Coxeter groups of coincidental types and type $D$ in terms of the corresponding Coxeter number of the group. In their paper, they also proposed multiple conjectures about enumerating elements with near maximal orbit length. We resolve all the conjectures that they have put forth about enumeration, and in the process we give complete classifications of these elements of Coxeter groups of types $A,B$ and $D$ with near maximal orbit lengths.
