Twisting the quantum grassmannian

Launois, S.; Lenagan, T.

doi:10.1090/S0002-9939-2010-10478-1

Twisting the quantum grassmannian
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by S. Launois and T. H. Lenagan

Proc. Amer. Math. Soc. 139 (2011), 99-110

DOI: https://doi.org/10.1090/S0002-9939-2010-10478-1

Published electronically: July 13, 2010

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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

Michael Artin, William Schelter, and John Tate, Quantum deformations of $\textrm {GL}_n$, Comm. Pure Appl. Math. 44 (1991), no. 8-9, 879–895. MR 1127037, DOI 10.1002/cpa.3160440804
Ken A. Brown and Ken R. Goodearl, Lectures on algebraic quantum groups, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser Verlag, Basel, 2002. MR 1898492, DOI 10.1007/978-3-0348-8205-7
K R Goodearl, S Launois and T H Lenagan, Totally nonnegative cells and matrix Poisson varieties, arXiv:0905.3631.
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.
K. R. Goodearl and T. H. Lenagan, Quantum determinantal ideals, Duke Math. J. 103 (2000), no. 1, 165–190. MR 1758243, DOI 10.1215/S0012-7094-00-10318-3
K. R. Goodearl and M. Yakimov, Poisson structures on affine spaces and flag varieties. II, Trans. Amer. Math. Soc. 361 (2009), no. 11, 5753–5780. MR 2529913, DOI 10.1090/S0002-9947-09-04654-6
A. C. Kelly, T. H. Lenagan, and L. Rigal, Ring theoretic properties of quantum Grassmannians, J. Algebra Appl. 3 (2004), no. 1, 9–30. MR 2047633, DOI 10.1142/S0219498804000630
Günter R. Krause and Thomas H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, Revised edition, Graduate Studies in Mathematics, vol. 22, American Mathematical Society, Providence, RI, 2000. MR 1721834, DOI 10.1090/gsm/022
Daniel Krob and Bernard Leclerc, Minor identities for quasi-determinants and quantum determinants, Comm. Math. Phys. 169 (1995), no. 1, 1–23. MR 1328259
A Knutson, T Lam and D E Speyer, Positroid varieties I: juggling and geometry, arXiv:0903.3694.
S. Launois, T. H. Lenagan, and L. Rigal, Prime ideals in the quantum Grassmannian, Selecta Math. (N.S.) 13 (2008), no. 4, 697–725. MR 2403308, DOI 10.1007/s00029-008-0054-z
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, DOI 10.1016/j.jalgebra.2005.10.021
T. H. Lenagan and L. Rigal, Quantum analogues of Schubert varieties in the Grassmannian, Glasg. Math. J. 50 (2008), no. 1, 55–70. MR 2381732, DOI 10.1017/S0017089507003928
T. H. Lenagan and E. J. Russell, Cyclic orders on the quantum Grassmannian, Arab. J. Sci. Eng. Sect. C Theme Issues 33 (2008), no. 2, 337–350 (English, with English and Arabic summaries). MR 2500045
A Postnikov, Total positivity, Grassmannians, and networks, arXiv:0609764.
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.