标题： 平凡的Kazhdan-Lusztig多项式和布劳特图的立方分解 显示英文标题

标题： Trivial Kazhdan-Lusztig polynomials and cubulation of the Bruhat graph

摘要： 对于任意的 Coxeter 系统$(W,S)$和任何$y \in W$，我们研究 Kazhdan-Lusztig 多项式$P_{x,y}$对所有$x \leq y$为平凡的条件，与区间$[1,y]$的 Bruhat 图可以被立方化的条件之间的关系，即这个图大致可以由$\mathbb{Z}$的子区间的乘积来覆盖。 在一方，我们结合Carrell-Peterson和Elias-Williamson的结果，证明如果$[1,y]$可以被立方化，那么$P_{x,y} = 1$对于所有$x \leq y$成立。 然后我们研究该陈述的逆命题。 对于$(W,S)$有限且$w_0$是$W$中的最长元素，使得$P_{x,w_0} = 1$对所有$x \in W$成立，我们构造了类型为$A$和$B/C$的$[1,w_0]$的立方分解。 然而，在某些特殊类型中，我们确定元素$y \in W$使得$P_{1,y} = 1$但$[1,y]$无法被立方化。 我们然后证明，如果存在无限多个 $y \in W$ 使得 $[1,y]$ 可以被立方化，那么 $(W,S)$ 必须是 $\tilde{A}_n$ 类型的，对于某些 $n \geq 1$。 最后，对于类型为$\tilde{A}_2$的$(W,S)$，我们为每个无限多个$y \in W$展示了$[1,y]$的立方分解，使得对于所有$x \leq y$都有$P_{x,y} = 1$。 显示英文摘要

摘要： For $(W,S)$ an arbitrary Coxeter system and any $y \in W$, we investigate the relationship between the condition that the Kazhdan-Lusztig polynomial $P_{x,y}$ is trivial for all $x \leq y$, and the condition that the Bruhat graph for the interval $[1,y]$ can be cubulated, meaning roughly that this graph can be spanned by a product of subintervals of $\mathbb{Z}$. In one direction, we combine results of Carrell-Peterson and Elias-Williamson to prove that if $[1,y]$ can be cubulated, then $P_{x,y} = 1$ for all $x \leq y$. We then investigate the converse of this statement. For $(W,S)$ finite and $w_0$ the longest element in $W$, so that $P_{x,w_0} = 1$ for all $x \in W$, we construct cubulations of $[1,w_0]$ in types $A$ and $B/C$. However, in some exceptional types, we determine elements $y \in W$ such that $P_{1,y} = 1$ but $[1,y]$ cannot be cubulated. We then prove that if there are infinitely many $y \in W$ such that $[1,y]$ can be cubulated, then $(W,S)$ must be of type $\tilde{A}_n$ for some $n \geq 1$. Finally, for $(W,S)$ of type $\tilde{A}_2$, we exhibit a cubulation of $[1,y]$ for each of the infinitely many $y \in W$ such that $P_{x,y} = 1$ for all $x \leq y$.