Duke Mathematical Journal

Tableau atoms and a new Macdonald positivity conjecture

L. Lapointe, A. Lascoux, and J. Morse

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Let $\lambda$ be the space of symmetric functions, and let $V\ sb k$ be the subspace spanned by the modified Schur functions $\{S\sb \lambda[X/(1-t)]\}\sb {\lambda\sb 1\leq k}$. We introduce a new family of symmetric polynomials, $\{A\sp {(k)}\sb \lambda[X;t]\}\sp {\lambda\sb 1\leq k}$, constructed from sums of tableaux using the charge statistic. We conjecture that the polynomials $A\sp {(k)}\sb \lambda[X;t]$ form a basis for $V\sb k$ and that the Macdonald polynomials indexed by partitions whose first part is not larger than $k$ expand positively in terms of our polynomials. A proof of this conjecture would not only imply the Macdonald positivity conjecture, but also substantially refine it. Our construction of the $A\sp {(k)}\sb \lambda[X;t]$ relies on the use of tableau combinatorics and yields various properties and conjectures on the nature of these polynomials. Another important development following from our investigation is that the $A\sp {(k)}\sb \lambda[X;t]$ seem to play the same role for $V\sb k$ as the Schur functions do for $\lambda$. In particular, this has led us to the discovery of many generalizations of properties held by the Schur functions, such as Pieri-type and Littlewood-Richardson-type coefficients.

Article information

Duke Math. J. Volume 116, Number 1 (2003), 103-146.

First available in Project Euclid: 26 May 2004

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Zentralblatt MATH identifier

Primary: 05E05: Symmetric functions and generalizations


Lapointe, L.; Lascoux, A.; Morse, J. Tableau atoms and a new Macdonald positivity conjecture. Duke Math. J. 116 (2003), no. 1, 103--146. doi:10.1215/S0012-7094-03-11614-2. http://projecteuclid.org/euclid.dmj/1085598237.

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  • W. Fulton, Young Tableaux: With Applications to Representation Theory and Geometry, London Math. Soc. Stud. Texts 35, Cambridge Univ. Press, Cambridge, 1997.
  • A. M. Garsia and M. Haiman, A graded representation module for Macdonald's polynomials, Proc. Natl. Acad. Sci. USA 90 (1993), 3607--3610.
  • A. M. Garsia and C. Procesi, On certain graded $S_n$-modules and the $q$-Kostka polynomials, Adv. Math. 94 (1992), 82--138.
  • M. Haiman, Hilbert schemes, polygraphs and the Macdonald positivity conjecture, J. Amer. Math Soc. 14 (2001), 941--1006.
  • N. H. Jing, Vertex operators and Hall-Littlewood symmetric functions, Adv. Math. 87 (1991), 226--248.
  • D. E. Knuth, Permutations, matrices, and generalized Young tableaux, Pacific J. Math. 34 (1970), 709--727.
  • L. Lapointe and J. Morse, Tableaux statistics for two part Macdonald polynomials, preprint.
  • A. Lascoux, B. Leclerc, and J.-Y. Thibon, Ribbon tableaux, Hall-Littlewood functions, quantum affine algebras, and unipotent varieties, J. Math. Phys. 38 (1997), 1041--1068.
  • --. --. --. --., ``The plactic monoid'' in Algebraic Combinatorics on Words by M. Lothaire, Encyclopedia Math. Appl. 90, Cambridge Univ. Press, Cambridge, 2002, 144--172. \CMP1 905 123
  • A. Lascoux and M.-P. Schützenberger, Sur une conjecture de H.O. Foulkes, C. R. Acad. Sci. Paris Sér. A-B 294 (1978), A323--A324.
  • --. --. --. --., ``Le mono\" \i de plaxique'' in Noncommutative Structures in Algebra and Geometric Combinatorics (Naples, 1978), Quad. ``Ricerca Sci.'' 109, Consiglio Nazionale delle Ricerche, Rome, 1981, 129--156.
  • I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2d ed., Oxford Math. Monogr., Oxford Univ. Press, New York, 1995.
  • G. de B. Robinson, On the representations of the symmetric group, Amer. J. Math. 60 (1938), 745--760.
  • C. Schensted, Longest increasing and decreasing subsequences, Canad. J. Math. 13 (1961), 179--191.
  • A. Schilling and S. O. Warnaar, Inhomogeneous lattice paths, generalized Kostka polynomials and $A_n-1$-supernomials, Comm. Math. Phys. 202 (1999), 359--401.
  • M. Shimozono, A cyclage poset structure for Littlewood-Richardson tableaux, European J. Combin. 22 (2001), 365--393.
  • --. --. --. --., Multi-atoms and monotonicity of generalized Kostka polynomials, European J. Combin. 22 (2001), 395--414.
  • M. Shimozono and J. Weyman, Graded characters of modules supported in the closure of a nilpotent conjugacy class, European J. Combin. 21 (2000), 257--288.
  • M. Shimozono and M. Zabrocki, Hall-Littlewood vertex operators and generalized Kostka polynomials, Adv. Math. 158 (2001), 66--85.
  • S. Veigneau, ACE: An algebraic combinatorics environment for the computer algebra system MAPLE, version 3.0, 1998, available from http://phalanstere.univ-mlv.fr/~ace/
  • M. A. Zabrocki, A Macdonald vertex operator and standard tableaux statistics for the two-column (q,t)-Kostka coefficients, Electron. J. Combin. 5 (1998), R45.