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Representation Theory
Representation Theory
ISSN 1088-4165

 

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


Authors: Antoine Mériaux and Gérard Cauchon
Journal: Represent. Theory 14 (2010), 645-687
MSC (2010): Primary 17B37; Secondary 16T20
Published electronically: November 1, 2010
MathSciNet review: 2736313
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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 $ \llbracket 1, \ldots,t \rrbracket$. 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$     $ \vert$     $ u$     $ \leq$     $ 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 \vert 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 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{$\Gamma$}}$-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.


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Additional Information

Antoine Mériaux
Affiliation: Laboratoire d’équations aux dérivées partielles et physique mathématique, U.F.R. Sciences, B.P. 1039, 51687 Reims Cedex 2, France.
Email: antoine.meriaux@univ-reims.fr

Gérard Cauchon
Affiliation: Laboratoire d’équations aux dérivées partielles et physique mathématique, U.F.R. Sciences, B.P. 1039, 51687 Reims Cedex 2, France
Email: gerard.cauchon@univ-reims.fr

DOI: http://dx.doi.org/10.1090/S1088-4165-2010-00382-9
PII: S 1088-4165(2010)00382-9
Received by editor(s): July 24, 2008
Received by editor(s) in revised form: March 24, 2010
Published electronically: November 1, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.