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Twisting the quantum grassmannian

Authors: S. Launois and T. H. Lenagan
Journal: Proc. Amer. Math. Soc. 139 (2011), 99-110
MSC (2010): Primary 16T20, 16P40, 16S38, 17B37, 20G42
Published electronically: July 13, 2010
MathSciNet review: 2729074
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Abstract: In contrast to the classical and semiclassical settings, the Coxeter element $ (12\dots n)$ which cycles the columns of an $ m\times n$ matrix does not determine an automorphism of the quantum grassmannian. Here, we show that this cycling can be obtained by means of a cocycle twist. A consequence is that the torus invariant prime ideals of the quantum grassmannian are permuted by the action of the Coxeter element $ (12\dots n)$. We view this as a quantum analogue of the recent result of Knutson, Lam and Speyer, where the Lusztig strata of the classical grassmannian are permuted by $ (12\dots n)$.

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

  • 1. M Artin, W Schelter and J Tate, Quantum deformations of $ {\rm GL}_n$, Comm. Pure Appl. Math. 44 (1991), no. 8-9, 879-895. MR 1127037 (92i:17014)
  • 2. 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)
  • 3. K R Goodearl, S Launois and T H Lenagan, Totally nonnegative cells and matrix Poisson varieties, arXiv:0905.3631.
  • 4. K R Goodearl, S Launois and T H Lenagan, Torus-invariant prime ideals in quantum matrices, totally nonnegative cells and symplectic leaves, arXiv:0909.3935. Mathematische Zeitschrift, doi:10.1007/s00209-010-0714-5.
  • 5. K R Goodearl and T H Lenagan, Quantum determinantal ideals, Duke Math. J. 103 (2000), 165-190. MR 1758243 (2001k:16080)
  • 6. K R Goodearl and M Yakimov, Poisson structures on affine spaces and flag varieties. II, Trans. Amer. Math. Soc. 361 (2009), 5753-5780. MR 2529913
  • 7. A Kelly, T H Lenagan, and L Rigal, Ring theoretic properties of quantum grassmannians, J. Algebra Appl. 3 (2004), no. 1, 9-30. MR 2047633 (2005b:20096)
  • 8. G R Krause and T H Lenagan, Growth of algebras and Gelfand-Kirillov dimension, Graduate Studies in Mathematics, 22. American Mathematical Society, Providence, RI, 2000. MR 1721834 (2000j:16035)
  • 9. D Krob and B Leclerc, Minor identities for quasi-determinants and quantum determinants, Comm. Math. Phys. 169 (1995), no. 1, 1-23. MR 1328259 (96g:15015)
  • 10. A Knutson, T Lam and D E Speyer, Positroid varieties I: juggling and geometry, arXiv:0903.3694.
  • 11. S Launois, T H Lenagan and L Rigal, Prime ideals in the quantum grassmannian, Selecta Mathematica 13 (2008), 697-725. MR 2403308 (2009e:20110)
  • 12. T H Lenagan and L Rigal, Quantum graded algebras with a straightening law and the AS-Cohen-Macaulay property for quantum determinantal rings and quantum grassmannians, J. Algebra 301 (2006), no. 2, 670-702. MR 2236763 (2007g:16059)
  • 13. T H Lenagan and L Rigal, Quantum analogues of Schubert varieties in the grassmannian, Glasgow Math. J. 50 (2008), no. 1, 55-70. MR 2381732 (2008m:20080)
  • 14. T H Lenagan and E J Russell, Cyclic orders on the quantum grassmannian, Arabian Journal for Science and Engineering 33 (2008), 337-350. MR 2500045
  • 15. A Postnikov, Total positivity, Grassmannians, and networks, arXiv:0609764.
  • 16. M Yakimov, Cyclicity of Lusztig's stratification of grassmannians and Poisson geometry, In: Noncommutative Structures in Mathematics and Physics, eds. S. Caenepeel, J. Fuchs, S. Gutt, Ch. Schweigert, A. Stolin, and F. van Oystaeyen, pp. 258-262, Royal Flemish Academy of Belgium for Sciences and Arts, 2010.

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Additional Information

S. Launois
Affiliation: School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury, Kent CT2 7NF, United Kingdom

T. H. Lenagan
Affiliation: Maxwell Institute for Mathematical Sciences, School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, King’s Buildings, Mayfield Road, Edinburgh EH9 3JZ, United Kingdom

Keywords: Quantum matrices, quantum grassmannian, cocycle twist, noncommutative dehomogenisation
Received by editor(s): October 1, 2009
Received by editor(s) in revised form: March 15, 2010
Published electronically: July 13, 2010
Additional Notes: The research of the first author was supported by a Marie Curie European Reintegration Grant within the $7^{th}$ European Community Framework Programme.
Communicated by: Martin Lorenz
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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