Remote Access Representation Theory
Green Open Access

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
DOI: https://doi.org/10.1090/S1088-4165-2010-00382-9
Published electronically: November 1, 2010
MathSciNet review: 2736313
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] J. Alev and F. Dumas, Sur le corps des fractions de certaines algèbres quantiques, J. Algebra 170 (1994), 229-265. MR 1302839 (96c:16033)
  • [2] N. Andruskiewitsch and F. Dumas, On the automorphisms of $ U_{q}^{+}(\mathfrak{g})$ ``Quantum Groups'' (B. Enriquez, ed.), IRMA Lectures in Mathematical and Theoretical Physics 12 European Math. Society (2008), 107-133. MR 2432991 (2009i:17019)
  • [3] K. A. Brown and K. R. Goodearl, Lectures on algebraic quantum groups, Advanced Courses in Mathematics, CRM Barcelona, Birkhäuser Verlag, Basel, 2002. MR 1898492 (2003f:16067)
  • [4] G. Cauchon, Effacement des dérivations et spectres premiers des algèbres quantiques, J. Algebra 260 (2003), 476-518. MR 1967309 (2004g:16044)
  • [5] G. Cauchon, Spectre premier de $ O_{q}(M_{n}(k)$: image canonique et séparation normale. J. Algebra 260 (2003), 419-569. MR 1967310 (2004g:16045)
  • [6] V. Deodhar, A combinatorial setting for questions in Kazhdan-Lusztig theory, Geometriae Dedicata 36 (1990), 95-119. MR 1065215 (91h:20075)
  • [7] K. R. Goodearl and E. S. Letzter, Prime factor algebras of the coordinate ring of quantum matrices, Proc. Amer. Math. Soc. 121 (1994), 1017-1025. MR 1211579 (94j:16066)
  • [8] K. R. Goodearl and R. B. Warfield Jr., An introduction to noncommutative nœtherian rings, London Math. Soc. Student texts 16, Cambridge University Press, Cambridge (1989). MR 1020298 (91c:16001)
  • [9] M. Gorelik, The prime and primitive spectra of a quantum Bruhat cell translate, J. Algebra 227 (2000), 211-253. MR 1754232 (2001d:17012)
  • [10] J. E. Humphreys, Introduction to Lie Algebras and Representations Theory, Graduate Texts in Mathematics 9, Springer-Verlag, New York (1980). MR 499562 (81b:17007)
  • [11] T. J. Hodges and T. Levasseur and M. Toro, Algebraic structure of multiparameter quantum groups, Advances in Maths. 126 (1997), 52-92. MR 1440253 (98e:17022)
  • [12] J. C. Jantzen, Lectures on quantum groups, Graduate Studies in Mathematics 6, Amer. Math. Soc., Providence (1996). MR 1359532 (96m:17029)
  • [13] A. Joseph, Quantum Groups and Their Primitive Ideals, Ergebnisse der Mathematik (3) 29, Springer-Verlag, Berlin (1995). MR 1315966 (96d:17015)
  • [14] S. Launois, Combinatorics of $ H$-primes in quantum matrices, J. Algebra 309 (2007), 139-167. MR 2301235 (2007k:05221)
  • [15] S. Launois, T. H. Lenagan and L. Rigal, Quantum unique factorisation domains, J. London Math. Soc. (2) 74 (2006), 321-340. MR 2269632 (2007h:16047)
  • [16] S. Z. Levendorskii and Ya. S. Soibelman, Algebras of functions on a compact quantum group, Schubert cells and quantum tori, Comm. Math. Phys. 139 (1991), 141-170. MR 1116413 (92h:58020)
  • [17] T. Lam and L. Williams, Total positivity for cominuscule grassmannians, New York Journal of Mathematics 14 (2008), 53-99. MR 2383586 (2008m:05307)
  • [18] R. Marsh and K. Rietsch, Parametrisations of flag varieties, Representation Theory, 8 (2004), 212-242. MR 2058727 (2005c:14061)
  • [19] A. Mériaux, Les diagrammes de Cauchon pour $ U_{q}^{+}(\mathfrak{g})$, J. Algebra 323 (2010), 1060-1097.
  • [20] A. Postnikov, Total positivity, Grassmannians and Networks, Preprint, 2006: http:// arxiv.org/abs/math/0609764.
  • [21] M. Yakimov, Invariant prime ideals in quantizations of nilpotent Lie algebras, Preprint, 2009: http://arxiv.org/abs/0905.0852
  • [22] H. Yamane, A Poincaré-Birkhoff-Witt theorem for quantized universal enveloping algebra of type $ A_{N},$ Publ. Res. Inst. Math. Sci. 25 (1989), 503-520. MR 1018513 (91a:17016)

Similar Articles

Retrieve articles in Representation Theory of the American Mathematical Society with MSC (2010): 17B37, 16T20

Retrieve articles in all journals with MSC (2010): 17B37, 16T20


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: https://doi.org/10.1090/S1088-4165-2010-00382-9
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.

American Mathematical Society