Skip to main content
SpringerLink
Log in
SpringerLink
Log in

Cluster Decorated Geometric Crystals, Generalized Geometric RSK-Correspondences, and Donaldson-Thomas Transformations

  • Chapter
  • First Online:
2017 MATRIX Annals

Part of the book series: MATRIX Book Series ((MXBS,volume 2))

  • 1138 Accesses

Abstract

For a simply connected, connected, semisimple complex algebraic group G, we define two geometric crystals on the \(\mathscr A\)-cluster variety of double Bruhat cell B −∩ Bw 0 B. These crystals are related by the ∗ duality. We define the graded Donaldson-Thomas correspondence as the crystal bijection between these crystals. We show that this correspondence is equal to the composition of the cluster chamber Ansatz, the inverse generalized geometric RSK-correspondence, and transposed twist map due to Berenstein and Zelevinsky.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 119.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 159.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Berenstein, A., Kazhdan, D.: Geometric and unipotent crystals II: from unipotent bicrystals to crystal bases. Contemporary Mathematics, vol. 433, pp. 13–88. American Mathematical Society, Providence (2007)

    Google Scholar 

  2. Berenstein, A., Zelevinsky, A.: Total positivity in Schubert varieties. Comment. Math. Helv. 72(1), 128–166 (1997)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. Berenstein, A., Fomin, S., Zelevinsky, A.: Parametrizations of canonical bases and totally positive matrices (with S. Fomin and A. Zelevinsky). Adv. Math. 122, 49–149 (1996)

    Google Scholar 

  5. Berenstein, A., Fomin, S., Zelevinsky, A.: Cluster algebras III: upper bounds and double Bruhat cells (2004). arXiv:math/0305434v3

    Google Scholar 

  6. Berenstein, A., Kirillov, A., Koshevoy, G.: Generalized RSK correspondences, Obervolfach reports, 23/2015, 1303–1305

    Google Scholar 

  7. Danilov, V., Karzanov, A., Koshevoy, G.: Tropical Plúcker functions and Kashiwara crystals. Contemporary Mathematics, vol. 616, pp. 77–100. American Mathematical Society, Providence (2014)

    Google Scholar 

  8. Fomin, S., Zelevinsky, A.: Cluster algebras I: foundations. J. Am. Math. Soc. 15, 497–529 (2002)

    Article  MathSciNet  Google Scholar 

  9. Genz, V., Koshevoy, G., Schumann, B.: Combinatorics of canonical bases revisited: type A (2017). arXiv:1611.03465

    Google Scholar 

  10. Genz, V., Koshevoy, G., Schumann, B.: Polyhedral parametrizations of canonical bases & cluster duality (2017). arXiv:1711.07176

    Google Scholar 

  11. Goncharov, A., Shen, L.: Donaldson-Thomas trasnsformations of moduli spaces of G-local systems (2016). arXiv:1602.06479

    Google Scholar 

  12. Gross, M., Hacking, P., Keel, S., Kontsevich, M.: Canonical bases for cluster algebras, preprint (2014). arXiv:1411.1394v2 [math.AG]

    Google Scholar 

  13. Kashiwara, M.: On crystal bases. In: Representations of Groups (Banff, AB, 1994). CMS Conference Proceedings, vol. 16, pp. 155–197. American Mathematical Society, Providence (1995)

    Google Scholar 

  14. Lee, K., Schiffler, R.: Positivity for cluster algebras. Ann. Math. 182, 73–125 (2015)

    Article  MathSciNet  Google Scholar 

  15. Lusztig, G.: Introduction to Quantum Groups. Progress in Mathematics, vol. 110. Birkháuser, Boston (1993)

    Google Scholar 

  16. Lusztig, G.: Total positivity in reductive groups. In: Lie Theory and Geometry. Progress in Mathematics, vol. 123, pp. 531–568. Birkháuser, Boston (1994)

    Google Scholar 

  17. Lusztig, G.: Piecewise linear parametrization of canonical bases (2008). arXiv:0807.2824

    Google Scholar 

  18. Nakashima, T.: Decorations on geometric crystals and monomial realizations of crystal bases for classical groups. J. Algebra 399, 712–769 (2014)

    Article  MathSciNet  Google Scholar 

  19. Nakashima, T., Zelevinsky, A.V.: Polyhedral realizations of crystal bases for quantized Kac-Moody algebras. Adv. Math. 131, 253–278 (1997)

    Article  MathSciNet  Google Scholar 

  20. Weng, D.: Donaldson-Thomas transformation of double Bruhat cells in semisimple lie groups (2016). arXiv:1611.04186

    Google Scholar 

Download references

Acknowledgements

I thank Arkady Berenstein, Volker Genz and Bea Schumann for inspired and fruitful discussions, organizers of the MATRIX workshop, and especially Paul Zinn-Justin, and the RSF grant 16-11-10075 for financial support.

Author information

Authors and Affiliations

  1. IITP RAS, MCCME, and Interdisciplinary Scientific Center J.-V. Poncelet, Moscow, Russia

    Gleb Koshevoy

Authors
  1. Gleb Koshevoy
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. The University of Melbourne, Melbourne, Australia

    Jan de Gier

  2. University of Western Australia, Perth, Australia

    Cheryl E. Praeger

  3. UCLA, Los Angeles, CA, USA

    Terence Tao

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Koshevoy, G. (2019). Cluster Decorated Geometric Crystals, Generalized Geometric RSK-Correspondences, and Donaldson-Thomas Transformations. In: de Gier, J., Praeger, C., Tao, T. (eds) 2017 MATRIX Annals. MATRIX Book Series, vol 2. Springer, Cham. https://doi.org/10.1007/978-3-030-04161-8_25

Download citation

Publish with us

Policies and ethics

Search

Navigation