Skip to main content
SpringerLink
Log in

Hook Formulas for Skew Shapes IV. Increasing Tableaux and Factorial Grothendieck Polynomials

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

We present a new family of hook-length formulas for the number of standard increasing tableaux which arise in the study of factorial Grothendieck polynomials. In the case of straight shapes, our formulas generalize the classical hook-length formula and the Littlewood formula. For skew shapes, our formulas generalize the Naruse hook-length formula and its q-analogs, which were studied in previous papers of the series.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. H. H. Andersen, J. C. Jantzen, and W. Soergel, Representations of Quantum Groups at pth Root of Unity and of Semisimple Groups in Characteristic p: Independence of p,” Astérisque, 220 (1994).

  2. S. Billey, “Kostant polynomials and the cohomology ring for G/B,” Duke Math. J., 96, 205–224 (1999).

    Article  MathSciNet  Google Scholar 

  3. M. Brion, “Lectures on the geometry of flag varieties,” in: Topics in Cohomological Studies of Algebraic Varieties, Birkh¨auser, Basel (2005), pp. 33–85.

    Chapter  Google Scholar 

  4. A. Buch, “A Littlewood–Richardson rule for the K-theory of Grassmannians,” Acta Math., 189, 37–78 (2002).

  5. A. Buch, “Combinatorial K-theory,” in: Topics in Cohomological Studies of Algebraic Varieties, Birkhäuser, Basel (2005), pp. 87–103.

  6. A. Buch, A. Kresch, M. Shimozono, H. Tamvakis, and A. Yong, “Stable Grothendieck polynomials and K-theoretic factor sequences,” Math. Ann., 340, 359–382 (2008).

    Article  MathSciNet  Google Scholar 

  7. D. Bump, P. J. McNamara, and M. Nakasuji, “Factorial Schur functions and the Yang–Baxter equation,” Comment. Math. Univ. St. Pauli, 63, 23–45 (2014).

  8. X. Chen and R. P. Stanley, “A formula for the specialization of skew Schur functions,” Ann. Comb., 20, 539–548 (2016).

    Article  MathSciNet  Google Scholar 

  9. I. Ciocan-Fontanine, M. Konvalinka, and I. Pak, “The weighted hook length formula,” J. Combin. Theory, Ser. A, 118, 1703–1717 (2011).

    Article  MathSciNet  Google Scholar 

  10. K. Dilks, O. Pechenik, and J. Striker, “Resonance in orbits of plane partitions and increasing tableaux,” J. Combin. Theory, Ser. A, 148, 244–274 (2017).

    Article  MathSciNet  Google Scholar 

  11. P. Edelman and C. Greene, “Balanced tableaux,” Adv. Math., 63, 42–99 (1987).

    Article  MathSciNet  Google Scholar 

  12. N. J. Y. Fan, P. L. Guo, and S. C. C. Sun, “Proof of a conjecture of Reiner–Tenner–Yong on barely set-valued tableaux,” SIAM J. Discrete Math., 33, 189–196 (2019).

    Article  MathSciNet  Google Scholar 

  13. S. Fomin and A. N. Kirillov, “Yang–Baxter equation, symmetric functions and Grothendieck polynomials,” preprint (1993); arXiv:hep-th/9306005.

  14. S. Fomin and A. N. Kirillov, “Grothendieck polynomials and the Yang–Baxter equation,” in: Proc. 6th FPSAC, DIMACS, Piscataway, New Jersey (1994), pp. 183–190.

  15. S. Fomin and A. N. Kirillov, “Reduced words and plane partitions,” J. Algebraic Combin., 6, 311–319 (1997).

    Article  MathSciNet  Google Scholar 

  16. J. S. Frame, G. de B. Robinson, and R. M. Thrall, “The hook graphs of the symmetric group,” Canad. J. Math., 6, 316–324 (1954).

  17. I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, New York (1983).

    MATH  Google Scholar 

  18. W. Graham and V. Kreiman, “Excited Young diagrams and equivariant K-theory, and Schubert varieties,” Trans. Amer. Math. Soc., 367, 6597–6645 (2015).

    Article  MathSciNet  Google Scholar 

  19. C. Greene, A. Nijenhuis, and H. S. Wilf, “A probabilistic proof of a formula for the number of Young tableaux of a given shape,” Adv. Math., 31, 104–109 (1979).

    Article  MathSciNet  Google Scholar 

  20. Z. Hamaker, R. Patrias, O. Pechenik, and N. Williams, “Doppelg¨angers: bijections of plane partitions,” Int. Math. Res. Not., 2020, No. 2, 487–540 (2020).

    Article  Google Scholar 

  21. Z. Hamaker, A. H. Morales, I. Pak, L. Serrano, and N. Williams, “Bijecting hidden symmetries for skew staircase shapes,” preprint (2021); arXiv:2103.09551.

  22. Z. Hamaker and B. Young, “Relating Edelman–Greene insertion to the Little map,” J. Algebraic Combin., 40, 693–710 (2014).

  23. B. H. Hwang, J. S. Kim, M. Yoo, and S. M. Yun, “Reverse plane partitions of skew staircase shapes and q-Euler numbers,” J. Combin. Theory, Ser. A, 168, 120–163 (2019).

    Article  MathSciNet  Google Scholar 

  24. T. Ikeda and H. Naruse, “Excited Young diagrams and equivariant Schubert calculus,” Trans. Amer. Math. Soc., 361, 5193–5221 (2009).

    Article  MathSciNet  Google Scholar 

  25. T. Ikeda and H. Naruse, “K-theoretic analogues of factorial Schur P- and Q-functions,” Adv. Math., 243, 22–66 (2013).

    Article  MathSciNet  Google Scholar 

  26. V. Kreiman, “Schubert classes in the equivariant K-theory and equivariant cohomology of the Grassmannian,” preprint (2005); arXiv:math.AG/0512204.

  27. A. Knutson and E. Miller, “Gröbner geometry of Schubert polynomials,” Ann. Math., 161, 1245–1318 (2005).

    Article  MathSciNet  Google Scholar 

  28. A. Knutson, E. Miller, and A. Yong, “Gr¨obner geometry of vertex decompositions and of flagged tableaux,” J. Reine Angew. Math., 630, 1–31 (2009).

    Article  MathSciNet  Google Scholar 

  29. M. Konvalinka, “A bijective proof of the hook-length formula for skew shapes,” European J. Combin., 88, 103104 (2020).

    Article  MathSciNet  Google Scholar 

  30. T. Lam, S. J. Lee, and M. Shimozono, “Back stable Schubert calculus,” Compos. Math., 157, 883–962 (2021).

    Article  MathSciNet  Google Scholar 

  31. A. Lascoux and M.-P. Schützenberger, “Polynômes de Schubert,” C. R. Acad. Sci. Paris, 294, No. 13, 447–450 (1982).

  32. A. Lascoux and M.-P. Schützenberger, “Structure de Hopf de l’anneau de cohomologie et de l’anneau de Grothendieck d’une vari´et´e de drapeaux,” C. R. Acad. Sci. Paris, 295, No. 11, 629–633 (1982).

  33. C. Lenart and A. Postnikov, “Affine Weyl groups in K-theory and representation theory,” Int. Math. Res. Not., 2007, No. 12, Art. ID rnm038 (2007).

  34. D. E. Littlewood, The Theory of Group Characters and Matrix Representations of Groups, 2nd edition, Oxford Univ. Press, New York (1950).

    MATH  Google Scholar 

  35. I. G. Macdonald, “Schur functions: theme and variations,” in: Publ. Inst. Rech. Math. Av., 498, Univ. Louis Pasteur, Strasbourg (1992), pp. 5–39.

  36. L. Manivel, Symmetric Functions, Schubert Polynomials and Degeneracy Loci, Amer. Math. Soc., Providence, Rhode Island (2001).

    MATH  Google Scholar 

  37. P. J. McNamara, “Factorial Grothendieck polynomials,” Electron. J. Combin., 13, No. 1, RP 71 (2006).

  38. P. J. McNamara, “Addendum to Factorial Grothendieck polynomials,” preprint (2011); available at tinyurl.com/3dfs7e9s.

  39. A. I. Molev and B. E. Sagan, “A Littlewood–Richardson rule for factorial Schur functions,” Trans. Amer. Math. Soc., 351, 4429–4443 (1999).

  40. C. Monical, B. Pankow, and A. Yong, “Reduced word enumeration, complexity, and randomization,” preprint (2019); arXiv:1901.03247.

  41. A. H. Morales, I. Pak, and G. Panova, “Hook formulas for skew shapes I. q-analogues and bijections,” J. Combin. Theory, Ser. A, 154, 350–405 (2018).

    Article  MathSciNet  Google Scholar 

  42. A. H. Morales, I. Pak, and G. Panova, “Hook formulas for skew shapes II. Combinatorial proofs and enumerative applications,” SIAM J. Discrete Math., 31, 1953–1989 (2017).

    Article  MathSciNet  Google Scholar 

  43. A. H. Morales, I. Pak, and G. Panova, “Hook formulas for skew shapes III. Multivariate and product formulas,” Algebr. Comb., 2, 815–861 (2019).

    MathSciNet  MATH  Google Scholar 

  44. A. H. Morales, I. Pak, and G. Panova, “Asymptotics of principal evaluations of Schubert polynomials for layered permutations,” Proc. Amer. Math. Soc., 147, 1377–1389 (2019).

    Article  MathSciNet  Google Scholar 

  45. A. H. Morales and D. G. Zhu, “On the Okounkov–Olshanski formula for standard tableaux of skew shapes,” preprint (2020); arXiv:2007.05006.

  46. H. Naruse, “Schubert calculus and hook formula,” talk slides at 73rd Sém. Lothar. Combin., Strobl, Austria (2014); available at tinyurl.com/z6paqzu.

  47. H. Naruse and S. Okada, “Skew hook formula for d-complete posets via equivariant K-theory,” Algebr. Comb., 2, 541–571 (2019).

    MathSciNet  MATH  Google Scholar 

  48. J.-C. Novelli, I. Pak, and A. V. Stoyanovskii, “A direct bijective proof of the hook-length formula,” Discrete Math. Theor. Comput. Sci., 1, 53–67 (1997).

    Article  MathSciNet  Google Scholar 

  49. A. Okounkov and G. Olshanski, “Shifted Schur functions,” St. Petersburg Math. J., 9, 239–300 (1998).

    MathSciNet  MATH  Google Scholar 

  50. I. Pak, “Hook length formula and geometric combinatorics,” S´em. Lothar. Combin., 46, Art. B46f (2001).

  51. I. Pak, “Skew shape asymptotics, a case-based introduction,” S´em. Lothar. Combin., 84, Art. B84a (2021).

  52. I. Pak and F. Petrov, “Hidden symmetries of weighted lozenge tilings,” Electron. J. Combin., 27, No. 3, #P3.44 (2020).

  53. O. Pechenik, “Cyclic sieving of increasing tableaux and small Schr¨oder paths,” J. Combin. Theory, Ser. A, 125, 357–378 (2014).

    Article  MathSciNet  Google Scholar 

  54. O. Pechenik, “Minuscule analogues of the plane partition periodicity conjecture of Cameron and Fon-Der-Flaass,” preprint (2021); arXiv:2107.02679.

  55. O. Pechenik and A. Yong, “Equivariant K-theory of Grassmannians,” Forum Math. Pi, 5, e3 (2017).

    Article  MathSciNet  Google Scholar 

  56. T. Pressey, A. Stokke, and T. Visentin, “Increasing tableaux, Narayana numbers and an instance of the cyclic sieving phenomenon,” Ann. Comb., 20, 609–621 (2016).

    Article  MathSciNet  Google Scholar 

  57. V. Reiner, B. Tenner, and A. Yong, “Poset edge densities, nearly reduced words, and barely set-valued tableaux,” J. Combin. Theory, Ser. A, 158, 66–125 (2018).

    Article  MathSciNet  Google Scholar 

  58. N. J. A. Sloane, The Online Encyclopedia of Integer Sequences, http://oeis.org.

  59. R. P. Stanley, Enumerative Combinatorics, Cambridge Univ. Press, Vol. 1, 2nd edition (2012) and Vol. 2 (1999).

  60. R. P. Stanley, “Some Schubert shenanigans,” preprint (2017); arXiv:1704.00851.

  61. R. A. Sulanke, “The Narayana distribution,” J. Statist. Plann. Inference, 101, 311–326 (2002).

    Article  MathSciNet  Google Scholar 

  62. H. Thomas and A. Yong, “A jeu de taquin theory for increasing tableaux, with applications to K-theoretic Schubert calculus,” Algebra Number Theory, 3, 121–148 (2009).

    Article  MathSciNet  Google Scholar 

  63. H. Thomas and A. Yong, “Longest increasing subsequences, Plancherel-type measure and the Hecke insertion algorithm,” Adv. Appl. Math., 46, 610–642 (2011).

    Article  MathSciNet  Google Scholar 

  64. H. Thomas and A. Yong, “Equivariant Schubert calculus and jeu de taquin,” Ann. Inst. Fourier (Grenoble), 68, 275–318 (2018).

    Article  MathSciNet  Google Scholar 

  65. A.Weigandt, “Bumpless pipe dreams and alternating sign matrices,” J. Combin. Theory, Ser. A, 182, Paper 105470 (2021).

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA, 01003, USA

    A. H. Morales

  2. Department of Mathematics, University of California, Los Angeles, CA, 90095, USA

    I. Pak

  3. Department of Mathematics, University of Southern California, Los Angeles, CA, 90089, USA

    G. Panova

Authors
  1. A. H. Morales
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. Pak
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. G. Panova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. H. Morales.

Additional information

Published in Zapiski Nauchnykh Seminarov POMI, Vol. 507, 2021, pp. 59–98.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Morales, A.H., Pak, I. & Panova, G. Hook Formulas for Skew Shapes IV. Increasing Tableaux and Factorial Grothendieck Polynomials. J Math Sci 261, 630–657 (2022). https://doi.org/10.1007/s10958-022-05777-0

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-022-05777-0

Search

Navigation