## Tokyo Journal of Mathematics

### Schur Coefficients of the Integral Form Macdonald Polynomials

Meesue YOO

#### Abstract

In this paper, we consider the combinatorial formula for the Schur coefficients of the integral form of the Macdonald polynomials. As an attempt to prove Haglund's conjecture that $\Biggl<\frac{J_{\mu}(X;q,q^k)}{(1-q)^{|\mu|}},s_{\lambda}(X)\Biggr> \in \mathbb{N}[q]$, we have found explicit combinatorial formulas for the Schur coefficients in one row case, two column case and certain hook shape cases [Yoo12]. A result of Egge-Loehr-Warrington [ELW] gives a combinatorial way of getting Schur expansion of symmetric functions when the expansion of the function in terms of Gessel's fundamental quasi symmetric functions is known. We apply this result to the combinatorial formula for the integral form Macdonald polynomials of Haglund [Hag] in quasi symmetric functions to prove the Haglund's conjecture in more general cases.

#### Article information

Source
Tokyo J. Math., Volume 38, Number 1 (2015), 153-173.

Dates
First available in Project Euclid: 21 July 2015

https://projecteuclid.org/euclid.tjm/1437506242

Digital Object Identifier
doi:10.3836/tjm/1437506242

Mathematical Reviews number (MathSciNet)
MR3374619

Zentralblatt MATH identifier
1256.05250

#### Citation

YOO, Meesue. Schur Coefficients of the Integral Form Macdonald Polynomials. Tokyo J. Math. 38 (2015), no. 1, 153--173. doi:10.3836/tjm/1437506242. https://projecteuclid.org/euclid.tjm/1437506242

#### References

• Sami Hayes Assaf, Dual equivalence graphs, ribbon tableaux and Macdonald polynomials, Thesis (Ph.D.)–University of California, Berkeley, ProQuest LLC, Ann Arbor, MI, 2007.
• Eric Egge, Nicholas A. Loehr and Gregory S. Warrington, From quasisymmetric expansions to Schur expansions via a modified inverse Kostka matrix, European J. Combin. 31 (2010), 2014–2027.
• Ömer Eğecioğlu and Jeffrey B. Remmel, A combinatorial interpretation of the inverse Kostka matrix, Linear and Multilinear Algebra 26 (1990), 59–84.
• Ira M. Gessel, Multipartite $P$-partitions and inner products of skew Schur functions, Combinatorics and algebra (Boulder, Colo., 1983), Contemp. Math. 34 (1984), 289–317.
• James Haglund, The $q$,$t$-Catalan numbers and the space of diagonal harmonics, University Lecture Series, vol. 41, American Mathematical Society, 2008.
• Mark Haiman, Macdonald polynomials and geometry, New perspectives in algebraic combinatorics (Berkeley, CA, 1996–97), Math. Sci. Res. Inst. Publ., Cambridge Univ. Press 38 (1999), 207–254.
• Mark Haiman, Hilbert schemes, polygraphs and the Macdonald positivity conjecture, J. Amer. Math. Soc. 14 (2001), no. 4, 941–1006 (electronic).
• James Haglund, Mark Haiman and Nick Loehr, A combinatorial formula for Macdonald polynomials, J. Amer. Math. Soc. 18 (2005), no. 3, 735–761 (electronic).
• Alain Lascoux and Marcel-Paul Schützenberger, Sur une conjecture de H. O. Foulkes, C. R. Acad. Sci. Paris Sér. A-B 286 (1978), no. 7, A323–A324.
• Ian. G. Macdonald, Symmetric functions and Hall polynomials, second ed., Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1995.
• Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999, With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin.
• Meesue Yoo, A combinatorial formula for the Schur coefficients of the integral form of the Macdonald polynomials in the two column and certain hook cases, Ann. Comb. 16 (2012), no. 2, 389–410.