Admissible diagrams in quantum nilpotent algebras and combinatoric properties of Weyl groups

Abstract:

Consider a complex simple Lie algebra $\mathfrak g$ of rank $n$. Denote by $\Pi$ a system of simple roots, by $W$ the corresponding Weyl group, consider a reduced expression $w = s_{\alpha _{1}} \circ \cdots \circ s_{\alpha _{t}}$ (each $\alpha _{i}\in \Pi$) of some $w \in W$ and call diagram any subset of $\lBrack 1, \ldots ,t \rBrack$. We denote by ${U^{+}[w]}$ (or $U_{q}^{w}(\mathfrak {g})$) the “quantum nilpotent” algebra as defined by Jantzen in 1996

We prove (theorem 5.3.1) that the positive diagrams naturally associated with the positive subexpressions (of the reduced expression of $w$ chosen above) in the sense of R. Marsh and K. Rietsch (or equivalently the subexpressions without defect in the sense of V. Deodhar), coincide with the admissible diagrams constructed by G. Cauchon which describe the natural stratification of $Spec({U^{+}[w]})$.

This theorem implies in particular (corollaries 5.3.1 and 5.3.2):

1. The map $\zeta : \Delta = \{j_{1}<\cdots < j_{s}\}\mapsto u = s_{\alpha _{j_{1}}}\circ \cdots \circ s_{\alpha _{j_{s}}}$ is a bijection from the set of admissible diagrams onto the set $\{u \in W \mbox { } | \mbox { } u \mbox { } \leq \mbox { } w \}$.

2. For each admissible diagram $\Delta = \{j_{1}<\cdots < j_{s}\}, s_{\alpha _{j_{1}}}\circ \cdots \circ s_{\alpha _{j_{s}}}$ is a reduced expression of $u= \zeta (\Delta )$.

3. The map $\zeta ^{\prime }: \Delta = \{j_{1}< \cdots < j_{s}\}\mapsto u^{\prime } = s_{\alpha _{j_{s}}}\circ \cdots \circ s_{\alpha _{j_{1}}}$ is a bijection from the set of admissible diagrams onto the set $\{u \in W | u \leq v = w^{-1} \}$.

4. For each admissible diagram $\Delta = \{j_{1}<\cdots <j_{s}\}, s_{\alpha _{j_{s}}}\circ \cdots \circ s_{\alpha _{j_{1}}}$ is a reduced expression of $u^{\prime } = \zeta ^{\prime } (\Delta )$.

If the Lie algebra $\mathfrak {g}$ is of type $A_{n}$ and $w$ is chosen in order that $U^+[w]$ is the algebra of quantum matrices $O_{q}(M_{p,m}(k))$ with $m = n-p+1$ (see section 2.1), then, the admissible diagrams are the $\hbox {\rotatedown {$Γ$}}$-diagrams in the sense of A. Postnikov (http://arxiv.org/abs/math/0609764). In this particular case, the assertions 3 and 4 have also been proved (with quite different methods) by A. Postnikov and by T. Lam and L. Williams.