GROSSBERG-KARSHON TWISTED CUBES AND HESITANT WALK AVOIDANCE

Abstract

Let G be a complex semisimple simply connected linear algebraic group. Let A be a dominant weight for G and I = (i(1), i(2), ... , i(n)) a word decomposition for an element w = s(i1), s(i2) ... s(in) 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 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, 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 lambda and 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 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 I. is hesitant-lambda-walk-avoiding. Our combinatorial condition imposes strong geometric conditions on the Bott Samelson variety associated to I.