Abstract
The goal of this work is to provide an elementary construction of the canonical basis \(\mathbf B(w)\) in each quantum Schubert cell \(U_q(w)\) and to establish its invariance under modified Lusztig’s symmetries. To that effect, we obtain a direct characterization of the upper global basis \(\mathbf {B}^{up}\) in terms of a suitable bilinear form and show that \(\mathbf B(w)\) is contained in \(\mathbf {B}^{up}\) and its large part is preserved by modified Lusztig’s symmetries.
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References
Beck, J., Chari, V., Pressley, A.: An algebraic characterization of the affine canonical basis. Duke Math. J. 99(3), 455–487 (1999)
Berenstein, A., Greenstein, J.: Double canonical bases. arXiv:1411.1391
Berenstein, A., Rupel, D.: Quantum cluster characters of Hall algebras. Sel. Math. (N.S.) 21(4), 1121–1176 (2015)
Berenstein, A., Zelevinsky, A.: String bases for quantum groups of type \(A_r\), I. M. Gel’fand Seminar. Advances in Soviet Mathematics, vol. 16, pp. 51–89. American Mathematical Society, Providence, RI (1993)
Berenstein, A., Zelevinsky, A.: Triangular bases in quantum cluster algebras. Int. Math. Res. Not. 2014(6), 1651–1688 (2014)
De Concini, C., Kac, V.G., Procesi, C.: Some quantum analogues of solvable Lie groups. Geometry and analysis (Bombay, 1992). Tata Inst. Fund. Res. Bombay, pp. 41–65 (1995)
Kimura, Y.: Quantum unipotent subgroup and dual canonical basis. Kyoto J. Math. 52(2), 277–331 (2012)
Kimura, Y.: Remarks on quantum unipotent subgroups and the dual canonical basis. Pac. J. Math. 286(1), 125–151 (2017)
Kimura, Y., Oya, H.: Quantum twists and dual canonical bases. arXiv:1604.07748
Kashiwara, M.: Global crystal bases of quantum groups. Duke Math. J. 69(2), 455–485 (1993)
Kassel, C.: Quantum Groups, Graduate Texts in Mathematics, vol. 155. Springer, New York (1995)
Kaveh, K., Khovanskii, A.G.: Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory. Ann. Math. (2) 176(2), 925–978 (2012)
Levendorskiĭ, S., Soibelman, Y.: Algebras of functions on compact quantum groups, Schubert cells and quantum tori. Commun. Math. Phys. 139(1), 141–170 (1991)
Lusztig, G.: Introduction to Quantum Groups, Progress in Mathematics, vol. 110. Birkhäuser, Boston (1993)
Lusztig, G.: Braid group action and canonical bases. Adv. Math. 122(2), 237–261 (1996)
Qin, F.: Compare triangular bases of acyclic quantum cluster algebras. arXiv:1606.05604
Tanisaki, T.: Modules over quantized coordinate algebras and PBW-bases. arXiv:1409.7973
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This work was partially supported by the NSF Grant DMS-1403527 (A. B.), by the Simons foundation collaboration Grant No. 245735 (J. G.), and by the ERC Grant MODFLAT and the NCCR SwissMAP of the Swiss National Science Foundation (A. B. and J. G.).
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Berenstein, A., Greenstein, J. Canonical bases of quantum Schubert cells and their symmetries. Sel. Math. New Ser. 23, 2755–2799 (2017). https://doi.org/10.1007/s00029-017-0316-8
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DOI: https://doi.org/10.1007/s00029-017-0316-8