Grossberg-Karshon twisted cubes and hesitant jumping walk avoidance

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 $\lambda$ 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.