Sign Types Associated to Posets

We start with a combinatorial definition of I-sign types which are a generalization of the sign types indexed by the root system of type Al (I ⊂ N finite). Then we study the set DI p of I-sign types associated to the partial orders on I. We establish a 1-1 correspondence between D [n] p and a certain set of convex simplexes in a euclidean space by which we get a geometric distinction of the sign types in D [n] p from the other [n]-sign types. We give a graph-theoretical criterion for an Sn-orbit O of D p to contain a dast and show that O contains at most one dast. Finally, we show the admirability of a poset associated to a dast. §0. Introduction. 0.1. Sign types indexed by the root system Φ of type Al were first introduced in the middle of the eighties for the description of the Kazhdan-Lusztig cells in the affine Weyl group Wa(Ãl) of type Ãl (see [8, 10]). Subsequently they were extended to the case where the root system Φ is of an arbitrary type (see [9]). These sign types were defined originally as the connected components of the complement in a euclidean space spanned by Φ after removing a certain set of hyperplanes, which are now known as admissible sign types. The cardinalities of the admissible sign