Cauchon diagrams for quantized enveloping algebras

Let g be a finite-dimensional complex simple Lie algebra, K a commutative field and q a nonzero element of K which is not a root of unity. To each reduced decomposition of the longest element w0 of the Weyl group W corresponds a PBW basis of the quantised enveloping algebra , and one can apply the theory of deleting-derivation to this iterated Ore extension. In particular, for each decomposition of w0, this theory constructs a bijection between the set of prime ideals in that are invariant under a natural torus action and certain combinatorial objects called Cauchon diagrams. In this paper, we give an algorithmic description of these Cauchon diagrams when the chosen reduced decomposition of w0 corresponds to a good ordering (in the sense of Lusztig (1990) [Lus90], ) of the set of positive roots. This algorithmic description is based on the constraints that are coming from Lusztig's admissible planes Lusztig (1990) [Lus90], : each admissible plane leads to a set of constraints that a diagram has to satisfy to be Cauchon. Moreover, we explicitly describe the set of Cauchon diagrams for explicit reduced decomposition of w0 in each possible type. In any case, we check that the number of Cauchon diagrams is always equal to the cardinal of W. In Cauchon and Mériaux (2008) [CM08], we use these results to prove that Cauchon diagrams correspond canonically to the positive subexpressions of w0. So the results of this paper also give an algorithmic description of the positive subexpressions of any reduced decomposition of w0 corresponding to a good ordering.