Twisting the quantum grassmannian
Authors:
S. Launois and T. H. Lenagan
Journal:
Proc. Amer. Math. Soc. 139 (2011), 99110
MSC (2010):
Primary 16T20, 16P40, 16S38, 17B37, 20G42
Published electronically:
July 13, 2010
MathSciNet review:
2729074
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Additional Information
Abstract: In contrast to the classical and semiclassical settings, the Coxeter element which cycles the columns of an 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 . 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 .
 1.
Michael
Artin, William
Schelter, and John
Tate, Quantum deformations of 𝐺𝐿_{𝑛},
Comm. Pure Appl. Math. 44 (1991), no. 89,
879–895. MR 1127037
(92i:17014), 10.1002/cpa.3160440804
 2.
Ken
A. Brown and Ken
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, Torusinvariant prime ideals in quantum matrices, totally nonnegative cells and symplectic leaves, arXiv:0909.3935. Mathematische Zeitschrift, doi:10.1007/s0020901007145.
 5.
K.
R. Goodearl and T.
H. Lenagan, Quantum determinantal ideals, Duke Math. J.
103 (2000), no. 1, 165–190. MR 1758243
(2001k:16080), 10.1215/S0012709400103183
 6.
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
(2010k:14092), 10.1090/S0002994709046546
 7.
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
(2005b:20096), 10.1142/S0219498804000630
 8.
Günter
R. Krause and Thomas
H. Lenagan, Growth of algebras and GelfandKirillov dimension,
Revised edition, Graduate Studies in Mathematics, vol. 22, American
Mathematical Society, Providence, RI, 2000. MR 1721834
(2000j:16035)
 9.
Daniel
Krob and Bernard
Leclerc, Minor identities for quasideterminants 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 Math.
(N.S.) 13 (2008), no. 4, 697–725. MR 2403308
(2009e:20110), 10.1007/s000290080054z
 12.
T.
H. Lenagan and L.
Rigal, Quantum graded algebras with a straightening law and the
ASCohenMacaulay property for quantum determinantal rings and quantum
Grassmannians, J. Algebra 301 (2006), no. 2,
670–702. MR 2236763
(2007g:16059), 10.1016/j.jalgebra.2005.10.021
 13.
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 (2008m:20080), 10.1017/S0017089507003928
 14.
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
(2010f:20048)
 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. 258262, Royal Flemish Academy of Belgium for Sciences and Arts, 2010.
 1.
 M Artin, W Schelter and J Tate, Quantum deformations of , Comm. Pure Appl. Math. 44 (1991), no. 89, 879895. 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, Torusinvariant prime ideals in quantum matrices, totally nonnegative cells and symplectic leaves, arXiv:0909.3935. Mathematische Zeitschrift, doi:10.1007/s0020901007145.
 5.
 K R Goodearl and T H Lenagan, Quantum determinantal ideals, Duke Math. J. 103 (2000), 165190. 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), 57535780. MR 2529913
 7.
 A Kelly, T H Lenagan, and L Rigal, Ring theoretic properties of quantum grassmannians, J. Algebra Appl. 3 (2004), no. 1, 930. MR 2047633 (2005b:20096)
 8.
 G R Krause and T H Lenagan, Growth of algebras and GelfandKirillov 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 quasideterminants and quantum determinants, Comm. Math. Phys. 169 (1995), no. 1, 123. 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), 697725. MR 2403308 (2009e:20110)
 12.
 T H Lenagan and L Rigal, Quantum graded algebras with a straightening law and the ASCohenMacaulay property for quantum determinantal rings and quantum grassmannians, J. Algebra 301 (2006), no. 2, 670702. 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, 5570. 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), 337350. 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. 258262, 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
Email:
S.Launois@kent.ac.uk
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
Email:
tom@maths.ed.ac.uk
DOI:
http://dx.doi.org/10.1090/S000299392010104781
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.
