Grossberg–Karshon twisted cubes and hesitant walk avoidance
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Let $G$ be a complex semisimple simply connected linear algebraic group. Let
$\lambda$ be a dominant weight for $G$ and $\mathcal{I} = (i_1, i_2, \ldots,
i_n)$ a word decomposition for an element $w = s_{i_1} s_{i_2} \cdots s_{i_n}$
of the Weyl group of $G$, where the $s_i$ are the simple reflections. In the
1990s, Grossberg and Karshon introduced a virtual lattice polytope associated
to $\lambda$ and $\mathcal{I}$, which they called a twisted cube, whose lattice
points encode (counted with sign according to a density function) characters of
representations of $G$. In recent work, the first author and Jihyeon Yang prove
that the Grossberg-Karshon twisted cube is untwisted (so the support of the
density function is a closed convex polytope) precisely when a certain
torus-invariant divisor on a toric variety, constructed from the data of
$\lambda$ and $\mathcal{I}$, is basepoint-free. This corresponds to the
situation in which the Grossberg-Karshon character formula is a true
combinatorial formula in the sense that there are no terms appearing with a
minus sign. In this note, we translate this toric-geometric condition to the
combinatorics of $\mathcal{I}$ and $\lambda$. More precisely, we introduce the
notion of hesitant $\lambda$-walks and then prove that the associated
Grossberg-Karshon twisted cube is untwisted precisely when $\mathcal{I}$ is
hesitant-$\lambda$-walk-avoiding.