Grossberg-Karshon扭曲立方体和犹豫跳跃步行回避
Grossberg-Karshon twisted cubes and hesitant jumping walk avoidance
摘要：
设$G$是秩为$r$的复单缀半单代数群，$B$是Borel子群.令$\mathbf i \in [r]^n$是一个字，令$\mathbf \ell = (\ell_1,\dots,\ell_n)$是一个非负整数序列。Grossberg和Karshon介绍了一种与$\mathbf i$和$\mathbf \ell$相关的虚格多面体，称为扭曲立方体，其格点编码$B$ -表示的特征。更精确地说，扭曲立方体中的格点，根据一定的密度函数用符号计数，产生由$\mathbf i$和$\mathbf \ell$确定的广义Demazure模的特征。在最近的工作中，作者和原田精确地描述了Grossberg-Karshon扭曲立方体何时解扭，即，在整数序列$\mathbf \ell$来自$G$的权重$λ$的情况下，扭曲立方体是闭凸多面体。然而，并非每个整数序列$\mathbf \ell$都来自$G$的权重。本文利用$\mathbf i$和$\mathbf \ell$的组合学，研究了任意词$\mathbf i$和任意整数序列$\mathbf \ell$对应的Grossberg-Karshon扭立方体的无扭性.事实上，我们证明了Grossberg-Karshon扭曲立方体是untwisted正是当$\mathbf i$是犹豫-跳跃- $\mathbf \ell$ -步行避免。
英文摘要：
Let $G$ be a complex simply-laced semisimple algebraic group of rank $r$ and $B$ a Borel subgroup. Let $\mathbf i \in [r]^n$ be a word and let $\mathbf \ell = (\ell_1,\dots,\ell_n)$ be a sequence of non-negative integers. Grossberg and Karshon introduced a virtual lattice polytope associated to $\mathbf i$ and $\mathbf \ell$ called a twisted cube, whose lattice points encode the character of a $B$-representation. More precisely, lattice points in the twisted cube, counted with sign according to a certain density function, yields the character of the generalized Demazure module determined by $\mathbf i$ and $\mathbf \ell$. In recent work, the author and Harada described precisely when the Grossberg-Karshon twisted cube is untwisted, i.e., the twisted cube is a closed convex polytope, in the situation when the integer sequence $\mathbf \ell$ comes from a weight $λ$ of $G$. However, not every integer sequence $\mathbf \ell$ comes from a weight of $G$. In the present paper, we interpret untwistedness of Grossberg-Karshon twisted cubes associated to any word $\mathbf i$ and any integer sequence $\mathbf \ell$ using the combinatorics of $\mathbf i$ and $\mathbf \ell$. Indeed, we prove that the Grossberg-Karshon twisted cube is untwisted precisely when $\mathbf i$ is hesitant-jumping-$\mathbf \ell$-walk-avoiding.
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