Journal of Algebraic Combinatorics

, Volume 33, Issue 2, pp 163–198 | Cite as

A computational and combinatorial exposé of plethystic calculus

Open Access


In recent years, plethystic calculus has emerged as a powerful technical tool for studying symmetric polynomials. In particular, some striking recent advances in the theory of Macdonald polynomials have relied heavily on plethystic computations. The main purpose of this article is to give a detailed explanation of a method for finding combinatorial interpretations of many commonly occurring plethystic expressions, which utilizes expansions in terms of quasisymmetric functions. To aid newcomers to plethysm, we also provide a self-contained exposition of the fundamental computational rules underlying plethystic calculus. Although these rules are well-known, their proofs can be difficult to extract from the literature. Our treatment emphasizes concrete calculations and the central role played by evaluation homomorphisms arising from the universal mapping property for polynomial rings.


Plethysm Symmetric functions Quasisymmetric functions LLT polynomials Macdonald polynomials 


  1. 1.
    Agaoka, Y.: An algorithm to calculate the plethysms of Schur functions. Mem. Fac. Integr. Arts Sci. Hiroshima Univ. IV 21, 1–17 (1995) Google Scholar
  2. 2.
    Atiyah, M.: Power operations in K-theory. Quart. J. Math. 17, 165–193 (1966) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Atiyah, M., Tall, D.: Group representations, λ-rings, and the J-homomorphism. Topology 8, 253–297 (1969) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bergeron, F., Bergeron, N., Garsia, A., Haiman, M., Tesler, G.: Lattice diagram polynomials and extended Pieri rules. Adv. Math. 2, 244–334 (1999) CrossRefMathSciNetGoogle Scholar
  5. 5.
    Bergeron, F., Garsia, A.: Science fiction and Macdonald polynomials. In: CRM Proceedings and Lecture Notes AMS VI, vol. 3, pp. 363–429 (1999) Google Scholar
  6. 6.
    Bergeron, F., Garsia, A., Haiman, M., Tesler, G.: Identities and positivity conjectures for some remarkable operators in the theory of symmetric functions. Methods Appl. Anal. VII 3, 363–420 (1999) MathSciNetGoogle Scholar
  7. 7.
    Bourbaki, N.: Elements of Mathematics: Algebra II. Springer, Berlin (1990), Chaps. 4–7 MATHGoogle Scholar
  8. 8.
    Carvalho, M., D’Agostino, S.: Plethysms of Schur functions and the shell model. J. Phys. A: Math. Gen. 34, 1375–1392 (2001) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Chen, Y., Garsia, A., Remmel, J.: Algorithms for plethysm. In: Greene, C. (ed.) Combinatorics and Algebra. Contemp. Math., vol. 34, pp. 109–153 (1984) Google Scholar
  10. 10.
    Duncan, D.: On D.E. Littlewood’s algebra of S-functions. Can. J. Math. 4, 504–512 (1952) MATHCrossRefGoogle Scholar
  11. 11.
    Foulkes, H.: Concomitants of the quintic and sextic up to degree four in the coefficients of the ground form. J. Lond Math. Soc. 25, 205–209 (1950) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Garsia, A.: Lecture notes from 1998 (private communication) Google Scholar
  13. 13.
    Garsia, A., Haglund, J.: A proof of the q,t-Catalan positivity conjecture. Discrete Math. 256, 677–717 (2002) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Garsia, A., Haglund, J.: A positivity result in the theory of Macdonald polynomials. Proc. Natl. Acad. Sci. 98, 4313–4316 (2001) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Garsia, A., Haiman, M.: A remarkable q,t-Catalan sequence and q-Lagrange inversion. J. Algebr. Comb. 5, 191–244 (1996) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Garsia, A., Haiman, M., Tesler, G.: Explicit plethystic formulas for Macdonald q,t-Kostka coefficients. In: Sém. Lothar. Combin., vol. 42, article B42m (1999), 45 pp. Google Scholar
  17. 17.
    Garsia, A., Remmel, J.: Plethystic formulas and positivity for q,t-Kostka coefficients. In: Progr. Math., vol. 161, pp. 245–262 (1998) Google Scholar
  18. 18.
    Garsia, A., Tesler, G.: Plethystic formulas for Macdonald q,t-Kostka coefficients. Adv. Math. 123, 144–222 (1996) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Geissenger, L.: Hopf Algebras of Symmetric Functions and Class Functions. Lecture Notes in Math., vol. 579. Springer, Berlin (1976) Google Scholar
  20. 20.
    Gessel, I.: Multipartite P-partitions and inner products of skew Schur functions. In: Combinatorics and Algebra (Boulder, Colo., 1983). Contemp. Math., vol. 34, pp. 289–317 (1984) Google Scholar
  21. 21.
    Grothendieck, A.: La theorie des classes de Chern. Bull. Soc. Math. Fr. 86, 137–154 (1958) MATHMathSciNetGoogle Scholar
  22. 22.
    Haglund, J.: A proof of the q,t-Schröder conjecture. Int. Math. Res. Not. 11, 525–560 (2004) CrossRefMathSciNetGoogle Scholar
  23. 23.
    Haglund, J., Haiman, M., Loehr, N.: A combinatorial formula for Macdonald polynomials. J. Am. Math. Soc. 18, 735–761 (2005) MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Haglund, J., Haiman, M., Loehr, N., Remmel, J., Ulyanov, A.: A combinatorial formula for the character of the diagonal coinvariants. Duke Math. J. 126, 195–232 (2005) MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Hoffman, P.: τ-rings and Wreath Product Representations. Lecture Notes in Math., vol. 706. Springer, Berlin (1979) MATHGoogle Scholar
  26. 26.
    Ibrahim, E.: On D.E. Littlewood’s algebra of S-functions. Proc. Am. Math. Soc. 7, 199–202 (1956) MATHMathSciNetGoogle Scholar
  27. 27.
    James, G., Kerber, A.: The Representation Theory of the Symmetric Group. Encyclopedia of Math. and Its Appl., vol. 16. Addison-Wesley, Reading (1981) MATHGoogle Scholar
  28. 28.
    Knutson, D.: λ-rings and the Representation Theory of the Symmetric Group. Lecture Notes in Math., vol. 308. Springer, Berlin (1976) Google Scholar
  29. 29.
    Lascoux, A.: Symmetric Functions & Combinatorial Operators on Polynomials. CBMS/AMS Lecture Notes, vol. 99 (2003) Google Scholar
  30. 30.
    Lascoux, A., Leclerc, B., Thibon, J.-Y.: Ribbon tableaux, Hall–Littlewood functions, quantum affine algebras, and unipotent varieties. J. Math. Phys. 38, 1041–1068 (1997) MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Littlewood, D.: Invariant theory, tensors and, group characters. Philos. Trans. R. Soc. A 239, 305–365 (1944) MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Littlewood, D.: The Theory of Group Characters, 2nd edn. Oxford University Press, London (1950) MATHGoogle Scholar
  33. 33.
    Macdonald, I.: A new class of symmetric functions. In: Actes du 20e Séminaire Lotharingien, vol. 372/S-20, pp. 131–171 (1988) Google Scholar
  34. 34.
    Macdonald, I.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, London (1995) MATHGoogle Scholar
  35. 35.
    Murnagham, F.: On the analyses of {m}{1k} and {m}{k}. Proc. Natl. Acad. Sci. 40, 721–723 (1954) CrossRefGoogle Scholar
  36. 36.
    Remmel, J.: The Combinatorics of Macdonald’s \(D_{n}^{1}\) operator. In: Sém. Lothar. Combin., article B54As (2006), 55 pp. Google Scholar
  37. 37.
    Sagan, B.: The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions. Wadsworth and Brooks/Cole, Belmont (1991) MATHGoogle Scholar
  38. 38.
    Stanley, R.: Enumerative Combinatorics, vol. 1. Cambridge University Press, Cambridge (1997) MATHGoogle Scholar
  39. 39.
    Stanley, R.: Enumerative Combinatorics, vol. 2. Cambridge University Press, Cambridge (1999) CrossRefGoogle Scholar
  40. 40.
    Uehara, H., Abotteen, E., Lee, M.-W.: Outer plethysms and λ-rings. Arch. Math. 46, 216–224 (1986) MATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Wybourne, B.: Symmetry Principles and Atomic Spectroscopy, with Appendix and Tables by P.H. Butler. Wiley, New York (1970) Google Scholar
  42. 42.
    Yang, M.: An algorithm for computing plethysm coefficients. Discrete Math. 180, 391–402 (1998) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2010

Authors and Affiliations

  1. 1.Virginia TechBlacksburgUSA
  2. 2.University of California, San DiegoLa JollaUSA

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