Shadows in Coxeter groups

Marius Graeber, Petra Schwer

Published 2018-07-23Version 1

For a given $w$ in a Coxeter group $W$ the elements $u$ smaller than $w$ in Bruhat order are the end-alcoves of stammering galleries of type $w$ in the Coxeter complex. We generalize this notion and consider sets of end-alcoves of galleries that are positively folded with respect to certain orientation $\phi$ of $\Sigma$. We call these sets \emph{shadows}. In this paper we will introduce various notions of orientations and shadows, study some of their properties and list some open questions.