Skip to main content
SpringerLink
Log in

PBW Degeneration: Algebra, Geometry, and Combinatorics

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

We review the recent progress in the theory of Poincaré–Birkhoff–Witt degenerations of irreducible representations of simple Lie algebras. We describe algebraic, geometric, and combinatorial aspects of the theory.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. I. Arzhantsev, “Flag varieties as equivariant compactifications of 𝔾n a ,” Proc. Am. Math. Soc., 139, No. 3, 783–786 (2011).

    Article  Google Scholar 

  2. I. Arzhantsev and E. Sharoiko, “Hassett–Tschinkel correspondence: modality and projective hypersurfaces,” J. Algebra, 348, No. 1, 217–232 (2011).

    Article  MathSciNet  Google Scholar 

  3. G. Andrews, The Theory of Partitions, Cambridge Univ. Press (1998).

    MATH  Google Scholar 

  4. T. Backhaus and D. Kus, “The PBW filtration and convex polytopes in type B,” arXiv:1504.06522.

  5. D. Barsky, Congruences pour les nombres de Genocchi de 2e espèce, Groupe d’étude d’Analyse ultramétrique, 8e année, no. 34, 1980/81.

  6. G. Cerulli Irelli and M. Lanini, “Degenerate flag varieties of type A and C are Schubert varieties,” Int. Math. Res. Not., No. 15, 6353–6374 (2015).

    Article  MathSciNet  Google Scholar 

  7. G. Cerulli Irelli, M. Lanini, and P. Littelmann, “Degenerate flag varieties and Schubert varieties: a characteristic free approach,” arxiv:1502.04590 (2015).

  8. P. Caldero, “Toric degenerations of Schubert varieties, Transform. Groups, 7, No. 1, 51–60 (2002).

    Article  MathSciNet  Google Scholar 

  9. I. Cherednik and E. Feigin, “Extremal part of the PBW-filtration and E-polynomials,” Adv. Math., 282, 220–264 (2015).

    Article  MathSciNet  Google Scholar 

  10. I. Cherednik and D. Orr, “Nonsymmetric difference Whittaker functions,” Math. Z., 279, No. 3, 879–938 (2015).

    Article  MathSciNet  Google Scholar 

  11. I. Cherednik and D. Orr, “One-dimensional nil-DAHA and Whittaker functions,” Transform, Groups, 18, No. 1, 23–59 (2013).

    Article  MathSciNet  Google Scholar 

  12. G. Cerulli Irelli, E. Feigin, and M. Reineke, “Quiver Grassmannians and degenerate flag varieties,” Algebra Number Theory, 6, No. 1, 165–194 (2012).

    Article  MathSciNet  Google Scholar 

  13. G. Cerulli Irelli, E. Feigin, and M. Reineke, “Degenerate flag varieties: moment graphs and Schröder numbers,” J. Alg. Combin., 38, No. 1, 159–189 (2013).

    Article  Google Scholar 

  14. G. Cerulli Irelli, E. Feigin, and M. Reineke, “Desingularization of quiver Grassmannians for Dynkin quivers,” Adv. Math., 245, 182–207 (2013).

    Article  MathSciNet  Google Scholar 

  15. G. Cerulli Irelli, E. Feigin, and M. Reineke, “Homological approach to the Hernandez–Leclerc construction and quiver varieties,” Representation Theory, 18, 1–14 (2014).

    Article  MathSciNet  Google Scholar 

  16. H. Dellac, “Problem 1735,” L’Intermédiaire des Mathématiciens, 7, 9–10 (1900).

    Google Scholar 

  17. D. Dumont, “Interprétations combinatoires des nombres de Genocchi,” Duke Math. J., 41, 305–318 (1974).

    Article  MathSciNet  Google Scholar 

  18. D. Dumont and A. Randrianarivony, “Dérangements et nombres de Genocchi,” Discr. Math., 132, 37–49 (1994).

    Article  Google Scholar 

  19. D. Dumont and G. Viennot, “A combinatorial interpretation of the Seidel generation of Genocchi numbers,” Discr. Math., 6, 77–87 (1980).

    MathSciNet  MATH  Google Scholar 

  20. D. Dumont and J. Zeng, “Further results on Euler and Genocchi numbers,” Aequat. Math., 47, 31–42 (1994).

    Article  MathSciNet  Google Scholar 

  21. E. Feigin, “The PBW filtration,” Represent. Theory, 13, 165–181 (2009).

    Article  MathSciNet  Google Scholar 

  22. E. Feigin, “The PBW filtration, Demazure modules and toroidal current algebras,” SIGMA, 4, 070 (2008).

    MathSciNet  MATH  Google Scholar 

  23. E. Feigin, “𝔾M a degeneration of flag varieties,” Select. Math., 18, No. 3, 513–537 (2012).

    Article  MathSciNet  Google Scholar 

  24. E. Feigin, “Degenerate flag varieties and the median Genocchi numbers,” Math. Res. Lett., 18, No. 6, 1–16 (2011).

    Article  MathSciNet  Google Scholar 

  25. E. Feigin, “The median Genocchi numbers, Q-analogues and continued fractions,” Eur. J. Combin., 33, 1913–1918 (2012).

    Article  MathSciNet  Google Scholar 

  26. E. Feigin, “Degenerate SL n: representations and flag varieties,” Funct. Anal. Appl., 48, No. 1, 59–71 (2014).

    Article  MathSciNet  Google Scholar 

  27. E. Feigin and M. Finkelberg, “Degenerate flag varieties of type A: Frobenius splitting and BWB theorem,” Math. Z., 275, Nos. 1–2, 55–77 (2013).

    Article  MathSciNet  Google Scholar 

  28. E. Feigin, M. Finkelberg, and P. Littelmann, “Symplectic degenerate flag varieties,” Can. J. Math., 66, No. 6, 1250–1286 (2014).

    Article  MathSciNet  Google Scholar 

  29. E. Feigin and I. Makedonskyi, “Nonsymmetric Macdonald polynomials, Demazure modules and PBW filtration,” J. Combin. Theory, Ser. A, 60–84 (2015).

  30. P. Flajolet, “Combinatorial aspects of continued fractions,” in: Proc. Colloq. Combinatorics-79, Univ. Montreal, 1979, Part II; Ann. Discr. Math., 9, 217–222 (1980).

  31. E. Feigin, G. Fourier, and P. Littelmann, “PBW-filtration and bases for irreducible modules in type A n,” Transform. Groups, 16, No. 1, 71–89 (2011).

    Article  MathSciNet  Google Scholar 

  32. E. Feigin, G. Fourier, and P. Littelmann, “PBW filtration and bases for symplectic Lie algebras,” Int. Math. Res. Not., 24, 5760–5784 (2011).

    Article  MathSciNet  Google Scholar 

  33. E. Feigin, G. Fourier, and P. Littelmann, “PBW-filtration over ℤ and compatible bases for V ℤ(λ) in type A n and C n,” Symmetries Integr. Syst. Represent., 40, 35–63 (2013).

    Article  Google Scholar 

  34. W. Fulton and J. Harris, Representation Theory. A First Course, Grad. Texts Math., 129, Springer-Verlag, New York (1991)

    Google Scholar 

  35. W. Fulton, Young Tableaux, with Applications to Representation Theory and Geometry, Cambridge Univ. Press, 1997.

  36. A. A. Gornitskii, “Essential signatures and canonical bases of irreducible representations of the group G 2,” Math. Notes, 97, No. 1, 30–41 (2015).

    Article  MathSciNet  Google Scholar 

  37. A. A. Gornitskii, “Essential signatures and canonical bases for irreducible representations of D 4,” arXiv:1507.07498.

  38. I. Gessel “Applications of the classical umbral calculus,” Alg. Univ., 49, 397–434 (2003).

    Article  MathSciNet  Google Scholar 

  39. N. Gonciulea and V. Lakshmibai, “Degenerations of flag and Schubert varieties to toric varieties,” Transform. Groups, 1, No. 3, 215–248(1996).

    Article  MathSciNet  Google Scholar 

  40. G.-N. Han and J. Zeng, “On a q-sequence that generalizes the median Genocchi numbers,” Ann. Sci. Math. Québec, 23, 63–72 (1999).

    MathSciNet  MATH  Google Scholar 

  41. G. Han and J. Zeng, “q-Polynomes de Gandhi et statistique de Denert,” Discr. Math., 205, Nos. 1-3, 119–143 (1999).

    Article  MathSciNet  Google Scholar 

  42. M. Haiman, J. Haglund, and N. Loehr, “A combinatorial formula for non-symmetric Macdonald polynomials,” Am. J. Math., 130, No. 2, 359–383 (2008).

    Article  Google Scholar 

  43. R. Hartshorne, Algebraic Geometry, Grad. Texts Math., 52, Springer-Verlag (1977).

  44. B. Hassett and Yu. Tschinkel, “Geometry of equivariant compactifications of 𝔾n a ,” Int. Math. Res. Not., 20, 1211–1230 (1999).

    Article  Google Scholar 

  45. A. Joseph, “On the Demazure character formula,” Ann. Sci. Ecole Norm. Super. (4), 18, No. 3, 389–419 (1985).

    Article  MathSciNet  Google Scholar 

  46. G. Kreweras, “Sur les permutations comptées par les nombres de Genocchi de 1-ière et 2-ième espèce,” Eur. J. Combin., 18, 49–58 (1997).

    Article  MathSciNet  Google Scholar 

  47. S. Kumar, Kac–Moody Groups, Their Flag Varieties and Representation Theory, Progr. Math., 204, Birkhäuser (2002).

  48. V. Lakshmibai, “Degenerations of flag varieties to toric varieties,” C. R. Acad. Sci. Paris, 321, 1229–1234 (1995).

    MathSciNet  MATH  Google Scholar 

  49. L. Seidel, “Über eine einfache Enstehungsweise der Bernoullischen Zahlen und einiger verwandten Reihen,” in: Münch. Akad. Wiss. Math. Phys. Cl. Sitzungsber, 4 (1877), pp. 157–187.

  50. N. J. A. Sloane, “Sequence A000366,” in: The On-Line Encyclopedia of Integer Sequences, http://oeis.org.

  51. E. Vinberg, “On some canonical bases of representation spaces of simple Lie algebras,” conference talk, Bielefeld (2005).

  52. X. G. Viennot, “Interprétations combinatoires des nombres d’Euler et Genocchi,” Sém. Th. Nombres, 1980/1981, No. 11.

  53. G. Viennot, “Interprétations combinatoires des nombres d’Euler et de Genocchi,” Sem. Number Theory, 1981/1982, No. 11, Univ. Bordeaux I, Talence (1982).

  54. G. Viennot, “Une théorie combinatoire des polynomes orthogonaux généraux,” in: Proc. Conf., Univ. du Quebec a Montreal (1983).

  55. J. Zeng and J. Zhou, “A q-analog of the Seidel generation of Genocchi numbers,” Eur. J. Combin., 27, 364–381 (2006).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. National Research University Higher School of Economics, Department of Mathematics, Moscow, Russia

    E. B. Feigin

Authors
  1. E. B. Feigin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. B. Feigin.

Additional information

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 136, Proceedings of the Seminar on Algebra and Geometry of Samara University, 2017

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Feigin, E.B. PBW Degeneration: Algebra, Geometry, and Combinatorics. J Math Sci 235, 685–713 (2018). https://doi.org/10.1007/s10958-018-4089-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-018-4089-1

Keywords and phrases

AMS Subject Classification

Search

Navigation