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

Abstract

Consider a complex simple Lie algebra g of rank n. Denote by Πa system of simple roots, by W the corresponding Weyl group, consider a reduced expression w = s_{α_{1}} ... s_{α_{t}} (each α_{i} in Π) 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.