Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Quasisymmetric expansions of Schur-function plethysms


Authors: Nicholas A. Loehr and Gregory S. Warrington
Journal: Proc. Amer. Math. Soc. 140 (2012), 1159-1171
MSC (2010): Primary 05E05, 05E10
DOI: https://doi.org/10.1090/S0002-9939-2011-10999-7
Published electronically: July 28, 2011
MathSciNet review: 2869102
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ s_{\mu}$ denote a Schur symmetric function and $ Q_{\alpha}$ a fundamental quasisymmetric function. Explicit combinatorial formulas are developed for the fundamental quasisymmetric expansions of the plethysms $ s_{\mu}[s_{\nu}]$ and $ s_{\mu}[Q_{\alpha}]$, as well as for related plethysms defined by inequality conditions. The key tools for obtaining these expansions are new standardization and reading word constructions for matrices.


References [Enhancements On Off] (What's this?)

  • 1. S. Assaf.
    Dual equivalence graphs. I: A combinatorial proof of LLT and Macdonald positivity.
    arXiv:1005.3759
  • 2. J. O. Carbonara, J. B. Remmel, and M. Yang.
    A combinatorial rule for the Schur function expansion of the plethysm $ s_{(1^a,b)}[p_k]$.
    Linear and Multilinear Algebra, 39(4):341-373, 1995. MR 1365453 (97b:05164)
  • 3. Luisa Carini and J. B. Remmel.
    Formulas for the expansion of the plethysms $ s_2[s_{(a,b)}]$ and $ s_2[s_{(n^k)}]$.
    Discrete Math., 193(1-3):147-177, 1998.
    Selected papers in honor of Adriano Garsia (Taormina, 1994). MR 1661367 (2000b:05129)
  • 4. C. Carré.
    Plethysm of elementary functions.
    Bayreuth. Math. Schr., (31):1-18, 1990. MR 1056146 (91f:20013)
  • 5. M. J. Carvalho and S. D'Agostino.
    Plethysms of Schur functions and the shell model.
    J. Phys. A, 34(7):1375-1392, 2001. MR 1819938 (2003a:05147)
  • 6. Y. M. Chen, A. M. Garsia, and J. Remmel.
    Algorithms for plethysm.
    In Combinatorics and algebra (Boulder, Colo., 1983), volume 34 of Contemp. Math., pages 109-153. Amer. Math. Soc., Providence, RI, 1984. MR 777698 (86f:05010)
  • 7. Ö. Egecioglu and J. B. Remmel. A combinatorial interpretation of the inverse Kostka matrix, Linear and Multilinear Algebra, 26(1-2):59-84, 1990. MR 1034417 (90m:05011)
  • 8. E. Egge, N. Loehr, and G. Warrington.
    From quasisymmetric expansions to Schur expansions via a modified inverse Kostka matrix.
    European J. Combin. 31(8):2014-2027, 2010. MR 2718279
  • 9. H. O. Foulkes.
    Concomitants of the quintic and sextic up to degree four in the coefficients of the ground form.
    J. London Math. Soc., 25:205-209, 1950. MR 0037276 (12:236e)
  • 10. W. Fulton.
    Young Tableaux; with applications to representation theory and geometry, volume 35 of London Mathematical Society Student Texts.
    Cambridge University Press, Cambridge, 1997. MR 1464693 (99f:05119)
  • 11. A. M. Garsia and J. Haglund.
    A proof of the $ q,t$-Catalan positivity conjecture.
    Discrete Math., 256(3):677-717, 2002.
    LaCIM 2000 Conference on Combinatorics, Computer Science and Applications (Montreal, QC). MR 1935784 (2004c:05207)
  • 12. Ira M. Gessel.
    Multipartite $ P$-partitions and inner products of skew Schur functions.
    In Combinatorics and algebra (Boulder, Colo., 1983), volume 34 of Contemp. Math., pages 289-317. Amer. Math. Soc., Providence, RI, 1984. MR 777705 (86k:05007)
  • 13. J. Haglund.
    A combinatorial model for the Macdonald polynomials.
    Proc. Natl. Acad. Sci. USA, 101(46):16127-16131 (electronic), 2004. MR 2114585 (2006e:05178)
  • 14. J. Haglund, M. Haiman, and N. Loehr.
    A combinatorial formula for Macdonald polynomials.
    J. Amer. Math. Soc., 102:2690-2696, 2005. MR 2141666 (2006g:05223b)
  • 15. T. M. Langley and J. B. Remmel.
    The plethysm $ s_\lambda[s_\mu]$ at hook and near-hook shapes.
    Electron. J. Combin., 11(1):Research Paper 11, 26 pp. (electronic), 2004. MR 2035305 (2004j:05128)
  • 16. Alain Lascoux, Bernard Leclerc, and Jean-Yves Thibon.
    Ribbon tableaux, Hall-Littlewood functions and unipotent varieties.
    Sém. Lothar. Combin., 34:Art. B34g, approx. 23 pp. (electronic), 1995. MR 1399754 (98m:05195)
  • 17. D. E. Littlewood.
    Invariant theory, tensors and group characters.
    Philos. Trans. Roy. Soc. London. Ser. A., 239:305-365, 1944. MR 0010594 (6:41c)
  • 18. N. Loehr and J. Remmel.
    A computational and combinatorial exposé of plethystic calculus.
    J. Algebraic Combin., 33(2):163-198, 2011. MR 2765321
  • 19. Nicholas A. Loehr and Gregory S. Warrington.
    Nested quantum Dyck paths and $ \nabla(s_\lambda)$.
    Int. Math. Res. Not. IMRN, (5):Art. ID rnm 157, 29, 2008. MR 2418288 (2009d:05257)
  • 20. I. G. Macdonald.
    Symmetric functions and Hall polynomials.
    Oxford Mathematical Monographs. Oxford University Press, New York, second edition, 1995.
    With contributions by A. Zelevinsky, Oxford Science Publications. MR 1354144 (96h:05207)
  • 21. Claudia Malvenuto and Christophe Reutenauer.
    Plethysm and conjugation of quasi-symmetric functions.
    Discrete Math., 193(1-3):225-233, 1998.
    Selected papers in honor of Adriano Garsia (Taormina, 1994). MR 1661370 (2000a:05211)
  • 22. S. P. O. Plunkett.
    On the plethysm of $ S$-functions.
    Canad. J. Math., 24:541-552, 1972. MR 0294526 (45:3596)
  • 23. R. M. Thrall.
    On symmetrized Kronecker powers and the structure of the free Lie ring.
    Amer. J. Math., 64:371-388, 1942. MR 0006149 (3:262d)
  • 24. Brian G. Wybourne.
    Symmetry principles and atomic spectroscopy.
    Wiley-Interscience [A division of John Wiley & Sons], New York-London-Sydney, 1970.
    Including an appendix of tables by P. H. Butler. MR 0421392 (54:9396)
  • 25. Mei Yang.
    An algorithm for computing plethysm coefficients.
    In Proceedings of the 7th Conference on Formal Power Series and Algebraic Combinatorics (Noisy-le-Grand, 1995), Discrete Math. 180:391-402, 1998. MR 1603696 (99d:05088)
  • 26. Mei Yang.
    The first term in the expansion of plethysm of Schur functions.
    Discrete Math., 246(1-3):331-341, 2002.
    Formal power series and algebraic combinatorics (Barcelona, 1999). MR 1887494 (2003e:05143)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 05E05, 05E10

Retrieve articles in all journals with MSC (2010): 05E05, 05E10


Additional Information

Nicholas A. Loehr
Affiliation: Department of Mathematics, Virginia Tech, Blacksburg, Virginia 24061
Email: nloehr@vt.edu

Gregory S. Warrington
Affiliation: Department of Mathematics and Statistics, University of Vermont, Burlington, Vermont 05401
Email: gwarring@cems.uvm.edu

DOI: https://doi.org/10.1090/S0002-9939-2011-10999-7
Received by editor(s): May 25, 2010
Received by editor(s) in revised form: December 24, 2010
Published electronically: July 28, 2011
Additional Notes: The first author was supported in part by National Security Agency grant H98230-08-1-0045
The second author was supported in part by National Security Agency grant H98230-09-1-0023
Communicated by: Jim Haglund
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society