Let
G
be a complex semisimple simply connected linear algebraic group. Let
λ be a dominant
weight for
G and
J=(i1,i2,…,in) a word decomposition
for an element
w=si1si2⋯sin of
the Weyl group of
G,
where the
si are
the simple reflections. In the 1990s, Grossberg and Karshon introduced a virtual lattice polytope
associated to
λ
and
J,
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, Harada and Jihyeon Yang proved 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
λ and
J,
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
J and
λ.
More precisely, we introduce the notion of
hesitantλ-walks and
then prove that the associated Grossberg–Karshon twisted cube is untwisted precisely
when
J is
hesitant-λ-walk-avoiding.
Our combinatorial condition imposes strong geometric conditions on the Bott–Samelson variety
associated to
J.
Keywords
Grossberg–Karshon twisted cubes, character formulae,
pattern avoidance