A computational and combinatorial exposé of plethystic calculus
Nicholas A. Loehr
and Jeffrey B. Remmel
DOI: 10.1007/s10801-010-0238-4
Abstract
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.
Pages: 163–198
Keywords: keywords plethysm; symmetric functions; quasisymmetric functions; LLT polynomials; Macdonald polynomials
Full Text: PDF
References
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2. Atiyah, M.: Power operations in K -theory. Quart. J. Math. 17, 165-193 (1966)
3. Atiyah, M., Tall, D.: Group representations, λ-rings, and the J -homomorphism. Topology 8, 253-297 (1969)
4. Bergeron, F., Bergeron, N., Garsia, A., Haiman, M., Tesler, G.: Lattice diagram polynomials and extended Pieri rules. Adv. Math. 2, 244-334 (1999)
5. Bergeron, F., Garsia, A.: Science fiction and Macdonald polynomials. In: CRM Proceedings and Lecture Notes AMS VI, vol. 3, pp. 363-429 (1999)
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)
7. Bourbaki, N.: Elements of Mathematics: Algebra II. Springer, Berlin (1990), Chaps. 4-7
8. Carvalho, M., D'Agostino, S.: Plethysms of Schur functions and the shell model. J. Phys. A: Math. Gen. 34, 1375-1392 (2001)
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)
10. Duncan, D.: On D.E. Littlewood's algebra of S-functions. Can. J. Math. 4, 504-512 (1952)
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)
12. Garsia, A.: Lecture notes from 1998 (private communication)
13. Garsia, A., Haglund, J.: A proof of the q, t -Catalan positivity conjecture. Discrete Math. 256, 677-717 (2002)
14. Garsia, A., Haglund, J.: A positivity result in the theory of Macdonald polynomials. Proc. Natl. Acad. Sci. 98, 4313-4316 (2001)
15. Garsia, A., Haiman, M.: A remarkable q, t -Catalan sequence and q-Lagrange inversion. J. Algebr. Comb. 5, 191-244 (1996)
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. J Algebr Comb (2011) 33: 163-198
17. Garsia, A., Remmel, J.: Plethystic formulas and positivity for q, t -Kostka coefficients. In: Progr. Math., vol. 161, pp. 245-262 (1998)
18. Garsia, A., Tesler, G.: Plethystic formulas for Macdonald q, t -Kostka coefficients. Adv. Math. 123, 144-222 (1996)
19. Geissenger, L.: Hopf Algebras of Symmetric Functions and Class Functions. Lecture Notes in Math., vol.
579. Springer, Berlin (1976)
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)
21. Grothendieck, A.: La theorie des classes de Chern. Bull. Soc. Math. Fr. 86, 137-154 (1958)
22. Haglund, J.: A proof of the q, t -Schröder conjecture. Int. Math. Res. Not. 11, 525-560 (2004)
23. Haglund, J., Haiman, M., Loehr, N.: A combinatorial formula for Macdonald polynomials. J. Am. Math. Soc. 18, 735-761 (2005)
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)
25. Hoffman, P.: τ-rings and Wreath Product Representations. Lecture Notes in Math., vol.
706. Springer, Berlin (1979)
26. Ibrahim, E.: On D.E. Littlewood's algebra of S-functions. Proc. Am. Math. Soc. 7, 199-202 (1956)
27. James, G., Kerber, A.: The Representation Theory of the Symmetric Group. Encyclopedia of Math. and Its Appl., vol.
16. Addison-Wesley, Reading (1981)
28. Knutson, D.: λ-rings and the Representation Theory of the Symmetric Group. Lecture Notes in Math., vol.
308. Springer, Berlin (1976)
29. Lascoux, A.: Symmetric Functions & Combinatorial Operators on Polynomials. CBMS/AMS Lecture Notes, vol. 99 (2003)
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)
31. Littlewood, D.: Invariant theory, tensors and, group characters. Philos. Trans. R. Soc. A 239, 305-365 (1944)
32. Littlewood, D.: The Theory of Group Characters, 2nd edn. Oxford University Press, London (1950)
33. Macdonald, I.: A new class of symmetric functions. In: Actes du 20e Séminaire Lotharingien, vol. 372/S-20, pp. 131-171 (1988)
34. Macdonald, I.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, London (1995)
35. Murnagham, F.: On the analyses of {m} \otimes {1k} and {m} \otimes {k}. Proc. Natl. Acad. Sci. 40, 721-723 (1954)
36. Remmel, J.: The Combinatorics of Macdonald's D1 n operator. In: Sém. Lothar. Combin., article B54As (2006), 55 pp.
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