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Shadows in Coxeter Groups

2020; Birkhäuser; Volume: 24; Issue: 1 Linguagem: Inglês

10.1007/s00026-019-00485-0

0219-3094

Marius Graeber, Petra Schwer,

Advanced Algebra and Geometry

For a given $w$ in a Coxeter group $W$ the elements $u$ smaller than $w$ in Bruhat order can be seen as the end-alcoves of stammering galleries of type $w$ in the Coxeter complex $\Sigma$. 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 shadows. Positively folded galleries are closely related to the geometric study of affine Deligne-Lusztig varieties, MV polytopes, Hall-Littlewood polynomials and many more agebraic structures. In this paper we will introduce various notions of orientations and hence shadows and study some of their algorithmic properties.