Grossberg-Karshon twisted cubes and hesitant jumping walk avoidance

Abstract

ⓒ The author. Let G be a complex simply-laced semisimple algebraic group of rank r and B a Borel subgroup. Let i is an element of [r](n) be a word and let l = (l(1), ...,l(n)) be a sequence of non-negative integers. Grossberg and Karshon introduced a virtual lattice polytope associated to i and l 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, yield the character of the generalized Dernazure module determined by i and l. In a 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 l comes from a weight lambda for G. However, not every integer sequence l comes from a weight for G. In the present paper, we interpret the untwistedness of Grossberg Karshon twisted cubes associated with any word i and any integer sequence t using the combinatorics of i and l. Indeed, we prove that the Grossberg-Karshon twisted cube is untwisted precisely when i is hesitant-jumping-l-walk-avoiding