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Affine PBW bases and MV polytopes in rank 2

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Abstract

Mirković–Vilonen (MV) polytopes have proven to be a useful tool in understanding and unifying many constructions of crystals for finite-type Kac-Moody algebras. These polytopes arise naturally in many places, including the affine Grassmannian, pre-projective algebras, PBW bases, and KLR algebras. There has recently been progress in extending this theory to the affine Kac-Moody algebras. A definition of MV polytopes in symmetric affine cases has been proposed using pre-projective algebras. In the rank-2 affine cases, a combinatorial definition has also been proposed. Additionally, the theory of PBW bases has been extended to affine cases, and, at least in rank-2, we show that this can also be used to define MV polytopes. The main result of this paper is that these three notions of MV polytope all agree in the relevant rank-2 cases. Our main tool is a new characterization of rank-2 affine MV polytopes.

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References

  1. Akasaka, T.: An integral PBW basis of the quantum affine algebra of type \(A^{(2)}_2\). Publ. Res. Inst. Math. Sci., 38, (2002). arXiv:math/0105170v1

  2. Anderson, J.: A polytope calculus for semisimple groups. Duke Math. J. 116, 567–588 (2003). arXiv:math/0110225

    Google Scholar 

  3. Baumann, P., Gaussent, S.: On Mirković-Vilonen cycles and crystal combinatorics. Represent. Theory, 12, 83–130 (2008). arXiv:math/0606711

    Google Scholar 

  4. Baumann, P., Dunlap, T., Kamnitzer, J.: Tingley Peter Rank 2 affine MV polytopes. arXiv:1202.6416

  5. Baumann, P., Kamnitzer, J.: Preprojective algebras and mv polytopes, to appear in Represent. Theory. arXiv:1009.2469

  6. Baumann, P., Kamnitzer, J., Tingley, P.: Affine Mirković-Vilonen polytopes. arXiv:1110.3661

  7. Beck, J.: Braid group action and quantum affine algebras. Commun. Math. Phys. 165 (1994). arXiv:hep-th/9404165

  8. Beck, J., Chari, V., Pressley, A.: An algebraic characterization of the affine canonical basis. Duke Math. J., 99, 455–487. arXiv:math/9808060 (1999)

    Google Scholar 

  9. Beck, J., Nakajima, H.: Crystal bases and two-sided cells of quantum affine algebras. Duke Math. J. 123, 335–402 (2004). arXiv:math/0212253

    Google Scholar 

  10. Berenstein, A., Zelevinsky, A.: Tensor product multiplicities, canonical bases, and totally positive varieties. Invent. Math. 143, 77–128. arXiv:math/9912012 (2001)

    Google Scholar 

  11. Braverman, A., Finkelberg, M., Gaitsgory, D.: Uhlenbeck spaces via affine Lie algebras. In: The Unity of Mathematics (volume dedicated to I. M. Gelfand’s 90th birthday). Progr. Math. vol. 244, pp. 17–135. Birkhäuser Boston, (2006). arXiv:math/0301176

  12. Buan, A.B., Iyama, O., Reiten, I., Scott, J.: Cluster structures for 2-Calabi-Yau categories and unipotent groups. Compos. Math. 145(4), 1035–1079 (2009). arXiv:math/0701557

    Google Scholar 

  13. Hong, J., Kang, S.-J.: Introduction to quantum groups and crystal bases. Graduate Studies in Mathematics, vol. 42, American Mathematical Society. Providence, RI (2002)

  14. Kac, Victor G.: Infinite-Dimensional Lie Algebras, 3rd edn. Cambridge University Press, Cambridge (1990)

    Book  MATH  Google Scholar 

  15. Kamnitzer, J.: Mirković-Vilonen cycles and polytopes. Ann. Math. 171(2), 731–777 (2010). arXiv:math/0501365

    Google Scholar 

  16. Kamnitzer, J.: The crystal structure on the set of Mirković-Vilonen polytopes. Adv. Math. 215, 66–93 (2007). arXiv:math/0505398

    Google Scholar 

  17. Kashiwara, M.: On crystal bases. CMS Conf. Proc., 16, (1995)

  18. Kashiwara, M., Saito, Y.: Geometric construction of crystal bases. Duke Math. J., 89(1), 9–36 (1997). arXiv:q-alg/9606009.

    Google Scholar 

  19. Kleshchev, A., Ram, A.: Representations of Khovanov–Lauda–Rouquier algebras and combinatorics of Lyndon words (1984), arXiv:0909.

  20. Lusztig, G.: Singularities, character formulas and a q-analog of weight multiplicities. Astérisque 101–102, 208–229 (1983)

    MathSciNet  Google Scholar 

  21. Lusztig, G.: Canonical bases arising from quantized enveloping algebras. J. Am. Math. Soc. 3, 447–498

  22. Lusztig, G.: Quivers, perverse sheaves, and quantized enveloping algebras. J. Am. Math. Soc. 4(2), 365–421 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  23. Lusztig, G.: Introduction to quantized universal enveloping algebras. Progr. in Math. vol. 105, Birkhäuser, (1992)

  24. Lusztig, G.: Piecewise linear parametrization of canonical bases. Pure Appl. Math. Q., 7 (2011). arXiv:0807.2824

  25. Muthiah, D.: Double MV Cycles and the Naito-Sagaki-Saito Crystal. arXiv:1108.5404

  26. Naito, S., Sagaki, D., Saito, Y.: Toward Berenstein–Zelevinsky data in affine type \(A\), I: construction of affine analogs. arXiv:1009.4526

  27. Saito, Y.: PBW basis of quantized universal enveloping algebras. Publ. Res. Inst. Math. Sci. 30(2), 209–232 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  28. Tingley, P., Webster, B.: Mirkovic–Vilonen polytopes and Khovanov–Lauda–Rouquier algebras. Preprint, arXiv:1210.6921v1

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Acknowledgments

We thank Pierre Baumann, Jonathan Beck, Alexander Braverman, Joel Kamnitzer and Hiraku Nakajima for many useful discussions.

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Authors and Affiliations

  1. Department of Mathematics, Brown University, Box 1917, Providence, RI, 02912, USA

    Dinakar Muthiah

  2. Department of Mathematics and Statistics, Loyola University Chicago, Chicago, IL, USA

    Peter Tingley

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  1. Dinakar Muthiah
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  2. Peter Tingley
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Correspondence to Dinakar Muthiah.

Additional information

The second author was supported by NSF grants DMS-0902649, DMS-1162385 and DMS-1265555.

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Muthiah, D., Tingley, P. Affine PBW bases and MV polytopes in rank 2. Sel. Math. New Ser. 20, 237–260 (2014). https://doi.org/10.1007/s00029-012-0117-z

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