Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-14T06:12:46.500Z Has data issue: false hasContentIssue false
  • English
  • Français

DUAL EQUIVALENCE GRAPHS I: A NEW PARADIGM FOR SCHUR POSITIVITY

Part of: Combinatorics Algebraic combinatorics

Published online by Cambridge University Press:  22 July 2015

SAMI H. ASSAF
SAMI H. ASSAF*
Affiliation:
Department of Mathematics, University of Southern California, Los Angeles, CA 90089-2532, USA; shassaf@usc.edu

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We make a systematic study of a new combinatorial construction called a dual equivalence graph. We axiomatize these graphs and prove that their generating functions are symmetric and Schur positive. This provides a universal method for establishing the symmetry and Schur positivity of quasisymmetric functions.

MSC classification

Primary: 05E05: Symmetric functions and generalizations
Secondary: 05A30: $q$-calculus and related topics 05E10: Combinatorial aspects of representation theory
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 3 , 2015 , e12
Check for updates
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author 2015

References

Assaf, S., ‘A generalized major index statistic’, Sém. Lothar. Combin. 60 (2008), Art. B50c; 13pp. (electronic).Google Scholar
Assaf, S. H., ‘Dual equivalence graphs, ribbon tableaux and Macdonald polynomials’, PhD Thesis, University of California Berkeley, 2007.Google Scholar
Carré, C. and Leclerc, B., ‘Splitting the square of a Schur function into its symmetric and antisymmetric parts’, J. Algebraic Combin. 4(3) (1995), 201–231.Google Scholar
Foata, D., ‘On the Netto inversion number of a sequence’, Proc. Amer. Math. Soc. 19 (1968), 236–240.Google Scholar
Gessel, I. M., ‘Multipartite P-partitions and inner products of skew Schur functions’, in: Combinatorics and Algebra (Boulder, CO, 1983), Contemporary Mathematics, 34 (American Mathematical Society, Providence, RI, 1984), 289–317.Google Scholar
Haglund, J., Haiman, M., Loehr, N., Remmel, J. B. and Ulyanov, A., ‘A combinatorial formula for the character of the diagonal coinvariants’, Duke Math. J. 126 (2005), 195–232.Google Scholar
Haiman, M. D., ‘Dual equivalence with applications, including a conjecture of Proctor’, Discrete Math. 99(1–3) (1992), 79–113.Google Scholar
Kashiwara, M., Miwa, T. and Stern, E., ‘Decomposition of q-deformed Fock spaces’, Selecta Math. (N.S.) 1(4) (1995), 787–805.Google Scholar
Lascoux, A., Leclerc, B. and Thibon, J.-Y., ‘Ribbon tableaux, Hall–Littlewood functions, quantum affine algebras, and unipotent varieties’, J. Math. Phys. 38(2) (1997), 1041–1068.CrossRefGoogle Scholar
Leclerc, B. and Thibon, J.-Y., ‘Littlewood–Richardson coefficients and Kazhdan–Lusztig polynomials’, in: Combinatorial Methods in Representation Theory (Kyoto, 1998), Advanced Studies in Pure Mathematics, 28 (Kinokuniya, Tokyo, 2000), 155–220.Google Scholar
Macdonald, I. G., Symmetric Functions and Hall Polynomials, 2nd edn, Oxford Mathematical Monographs (The Clarendon Press, Oxford University Press, New York, 1995). With contributions by A. Zelevinsky, Oxford Science Publications.Google Scholar
Roberts, A., ‘Dual equivalence graphs revisited and the explicit Schur expansion of a family of LLT polynomials’, J. Algebraic Combin. 39(2) (2014), 389–428.Google Scholar
Stanton, D. W. and White, D. E., ‘A Schensted algorithm for rim hook tableaux’, J. Combin. Theory Ser. A 40(2) (1985), 211–247.Google Scholar
van Leeuwen, M. A. A., ‘Spin-preserving Knuth correspondences for ribbon tableaux’, Electron. J. Combin. 12 (2005), Research Paper 10; 65pp. (electronic).Google Scholar