Grossberg–Karshon twisted cubes and hesitant walk avoidance

Megumi Harada and Eunjeong Lee

Vol. 278 (2015), No. 1, 119–136

DOI: 10.2140/pjm.2015.278.119

Abstract

Let $G$ be a complex semisimple simply connected linear algebraic group. Let $\lambda$ be a dominant weight for $G$ and $\mathcal{ℐ}=\left(i_1,i_2,\dots ,i_n\right)$ 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, 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 $\mathcal{ℐ}$, 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. Our combinatorial condition imposes strong geometric conditions on the Bott–Samelson variety associated to $\mathcal{ℐ}$.

Keywords

Grossberg–Karshon twisted cubes, character formulae, pattern avoidance

Mathematical Subject Classification 2010

Primary: 20G05

Secondary: 52B20

Milestones

Received: 3 October 2014

Revised: 3 March 2015

Accepted: 7 April 2015

Published: 30 September 2015