2009/02/04 by Antoine Mériaux, Mériaux, Antoine, G. Cauchon +2 · 3 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Random Matrices and Applications #math.QA #math.RT #msc:16W35 #msc:17B37
paper · pdf · doi:10.48550/arxiv.0902.0754
arxiv created 2009/02/04 · arxiv updated 2009/12/01
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_α1 ... s_αt (each αi in Π) of some w ∈ W and call diagram any subset of 1, ..., t. We denote by Uqw(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(Uqw(g)). If the Lie algebra g is of type An and w is choosen in order that Uqw(g) is the quantum matrices algebra Oq(Mp,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.