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Consider a complex simple Lie algebra g of rank n. Denote by \Pi a system ofsimple roots, by W the corresponding Weyl group, consider a reduced expressionw = s_{\alpha_{1}} ... s_{\alpha_{t}} (each \alpha_{i} in \Pi) of some w \in Wand call diagram any subset of {1, ..., t}. We denote by U_{q}^{w}(g) the"quantum nilpotent" algebra defined by J. C. Jantzen. We prove (theorem 5.3. 1)that the positive diagrams naturally associated with the positivesubexpressions (of the reduced expression of w) in the sense of R. Marsh and K.Rietsch, coincide with the admissible diagrams constructed by G. Cauchon whichdescribe the natural stratification of Spec(U_{q}^{w}(g)). If the Lie algebra gis of type A_{n} and w is choosen in order that U_{q}^{w}(g) is the quantummatrices algebra O_{q}(M_{p,m}(k)) with m = n-p+1 (see section 2.1), then theadmissible diagrams are known (G. Cauchon) to be the Le - diagrams in the senseof A. Postnikov . In this particular case, the equality of Le - diagrams andpositive subexpressions (of the reduced expression of w) have also been proved(with quite different methods) by A. Postnikov and by T. Lam and L. Williams.

Admissible Diagrams in U_{q}^{w}(g) and Combinatoric Properties of Weyl Groups