  Consider a complex simple Lie algebra g of rank n. Denote by \Pi a system of
simple roots, by W the corresponding Weyl group, consider a reduced expression
w = s_{\alpha_{1}} ... s_{\alpha_{t}} (each \alpha_{i} in \Pi) of some w \in W
and 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 positive
subexpressions (of the reduced expression of w) in the sense of R. Marsh and K.
Rietsch, coincide with the admissible diagrams constructed by G. Cauchon which
describe the natural stratification of Spec(U_{q}^{w}(g)). If the Lie algebra g
is of type A_{n} and w is choosen in order that U_{q}^{w}(g) is the quantum
matrices algebra O_{q}(M_{p,m}(k)) with m = n-p+1 (see section 2.1), then the
admissible diagrams are known (G. Cauchon) to be the Le - diagrams in the sense
of A. Postnikov . In this particular case, the equality of Le - diagrams and
positive 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.
