A combinatorial formula for the character of the diagonal coinvariants

previous :: next

A combinatorial formula for the character of the diagonal coinvariants

J. Haglund, M. Haiman, N. Loehr, J. B. Remmel, and A. Ulyanov

Source: Duke Math. J. Volume 126, Number 2 (2005), 195-232.

Abstract

Let R_n be the ring of coinvariants for the diagonal action of the symmetric group S_n. It is known that the character of R_n as a doubly graded S_n-module can be expressed using the Frobenius characteristic map as $\nabla e_{n}$ , where e_n is the nth elementary symmetric function and $\nabla$ is an operator from the theory of Macdonald polynomials. We conjecture a combinatorial formula for $\nabla e_{n}$ and prove that it has many desirable properties that support our conjecture. In particular, we prove that our formula is a symmetric function (which is not obvious) and that it is Schur positive. These results make use of the theory of ribbon tableau generating functions of Lascoux, Leclerc, and Thibon. We also show that a variety of earlier conjectures and theorems on $\nabla e_{n}$ are special cases of our conjecture.

Finally, we extend our conjectures on $\nabla e_{n}$ and several of the results supporting them to higher powers $\nabla^{m}e_{n}$ .

Primary Subjects: 05E10

Secondary Subjects: 05A30, 20C30

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.

If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.dmj/1106332719
Digital Object Identifier: doi:10.1215/S0012-7094-04-12621-1
Mathematical Reviews number (MathSciNet): MR2115257
Zentralblatt MATH identifier: 1069.05077

back to Table of Contents

References

F. Bergeron, A. M. Garsia, M. Haiman, and G. Tesler, Identities and positivity conjectures for some remarkable operators in the theory of symmetric functions, Methods Appl. Anal. 6 (1999), 363--420.

Mathematical Reviews (MathSciNet): MR1803316

E. S. Egge, J. Haglund, K. Killpatrick, and D. Kremer, A Schröder generalization of Haglund's statistic on Catalan paths, Electron. J. Combin. 10 (2003), research paper 16.

Mathematical Reviews (MathSciNet): MR1975766

A. M. Garsia and J. Haglund, A positivity result in the theory of Macdonald polynomials, Proc. Natl. Acad. Sci. USA 98 (2001), 4313--4316.

Mathematical Reviews (MathSciNet): MR1819133

Digital Object Identifier: doi:10.1073/pnas.071043398

--. --. --. --., ``A proof of the $q,t$-Catalan positivity conjecture'' in LaCIM 2000 Conference on Combinatorics, Computer Science and Applications (Montreal), Discrete Math. 256, Elsevier, Amsterdam, 2002, 677--717.

Mathematical Reviews (MathSciNet): MR1935784

Digital Object Identifier: doi:10.1016/S0012-365X(02)00343-6

A. M. Garsia and M. Haiman, A remarkable $q,t$-Catalan sequence and $q$-Lagrange inversion, J. Algebraic Combin. 5 (1996), 191--244.

Mathematical Reviews (MathSciNet): MR1394305

Digital Object Identifier: doi:10.1023/A:1022476211638

I. M. Gessel, ``Multipartite $P$-partitions and inner products of skew Schur functions'' in Combinatorics and Algebra (Boulder, Colo., 1983), Contemp. Math. 34, Amer. Math. Soc., Providence, 1984, 289--317.

Mathematical Reviews (MathSciNet): MR0777705

J. Haglund, Conjectured statistics for the $q,t$-Catalan numbers, Adv. Math. 175 (2003), 319--334.

Mathematical Reviews (MathSciNet): MR1972636

Digital Object Identifier: doi:10.1016/S0001-8708(02)00061-0

--. --. --. --., A proof of the $q,t$-Schröder conjecture, Int. Math. Res. Not. 2004, no. 11, 525--560.

Mathematical Reviews (MathSciNet): MR2038776

Digital Object Identifier: doi:10.1155/S1073792804132509

J. Haglund and N. Loehr, ``A conjectured combinatorial formula for the Hilbert series for diagonal harmonics'' to appear in Formal Power Series and Algebraic Combinatorics (Melbourne, Australia, 2002), http://www.math.upenn.edu/$\tilde\ $jhaglund/research.html

Mathematical Reviews (MathSciNet): MR2163448

Digital Object Identifier: doi:10.1016/j.disc.2004.01.022

Mark Haiman, Conjectures on the quotient ring by diagonal invariants, J. Algebraic Combin. 3 (1994), 17--76.

Mathematical Reviews (MathSciNet): MR1256101

Digital Object Identifier: doi:10.1023/A:1022450120589

--. --. --. --., ``$t,q$-Catalan numbers and the Hilbert scheme'' in Selected Papers in Honor of Adriano Garsia (Taormina, Italy, 1994), Discrete Math. 193, Elsevier, Amsterdam, 1998, 201--224.

Mathematical Reviews (MathSciNet): MR1661369

Digital Object Identifier: doi:10.1016/S0012-365X(98)00141-1

--. --. --. --., Hilbert schemes, polygraphs and the Macdonald positivity conjecture, J. Amer. Math. Soc. 14 (2001), 941--1006.

Mathematical Reviews (MathSciNet): MR1839919

Digital Object Identifier: doi:10.1090/S0894-0347-01-00373-3

JSTOR: links.jstor.org

--. --. --. --., Vanishing theorems and character formulas for the Hilbert scheme of points in the plane, Invent. Math. 149 (2002), 371--407.

Mathematical Reviews (MathSciNet): MR1918676

Digital Object Identifier: doi:10.1007/s002220200219

--. --. --. --., ``Combinatorics, symmetric functions and Hilbert schemes'' in Current Developments in Mathematics (Cambridge, Mass., 2002), International Press, Somerville, Mass., 2003, 39--112.

Mathematical Reviews (MathSciNet): MR2051783

G. James and A. Kerber, The Representation Theory of the Symmetric Group, Encyclopedia Math. Appl. 16, Addison-Wesley, Reading, Mass., 1981.

Mathematical Reviews (MathSciNet): MR0644144

M. Kashiwara and T. Tanisaki, Parabolic Kazhdan-Lusztig polynomials and Schubert varieties, J. Algebra 249 (2002), 306--325.

Mathematical Reviews (MathSciNet): MR1901161

Digital Object Identifier: doi:10.1006/jabr.2000.8690

S. V. Kerov, A. N. Kirillov, and N. Yu. Reshetikhin, Combinatorics, the Bethe ansatz and representations of the symmetric group (in Russian), Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 155 (1986), 50--64.; English translation in J. Soviet Math. 41 (1988), 916--924.

Mathematical Reviews (MathSciNet): MR0869576

A. N. Kirillov and N. Yu. Reshetikhin, The Bethe ansatz and the combinatorics of Young tableaux (in Russian), Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 155 (1986), 65--115.; English translation in J. Soviet Math. 41 (1988), 925--955.

Mathematical Reviews (MathSciNet): MR0869577

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.

Mathematical Reviews (MathSciNet): MR1434225

Digital Object Identifier: doi:10.1063/1.531807

B. Leclerc and J.-Y. Thibon, ``Littlewood-Richardson coefficients and Kazhdan-Lusztig polynomials'' in Combinatorial Methods in Representation Theory (Kyoto, 1998), Adv. Stud. Pure Math. 28, Kinokuniya, Tokyo, 2000, 155--220.

Mathematical Reviews (MathSciNet): MR1864481

N. Loehr, Conjectured statistics for the higher $q,t$-Catalan sequences, preprint, 2003, http://www.math.upenn.edu/$\tilde\ $nloehr/mcat.pdf

Mathematical Reviews (MathSciNet): MR2134172

--------, Multivariate analogues of Catalan numbers, parking functions, and their extensions, Ph.D. dissertation, University of California, San Diego, San Diego, 2003, http://math.ucsd.edu/$\tilde\ $thesis/year/2003.html

N. A. Loehr and J. B. Remmel, Conjectured combinatorial models for the Hilbert series of generalized diagonal harmonics modules, Electron. J. Combin. 11 (2004), no. R68.

Mathematical Reviews (MathSciNet): MR2097334

I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., Oxford Math. Monogr., Oxford Univ. Press, New York, 1995.

Mathematical Reviews (MathSciNet): MR1354144

I. Pak, Ribbon tile invariants, Trans. Amer. Math. Soc. 352 (2000), 5525--5561.

Mathematical Reviews (MathSciNet): MR1781275

Digital Object Identifier: doi:10.1090/S0002-9947-00-02666-0

JSTOR: links.jstor.org

A. Schilling, M. Shimozono, and D. E. White, ``Branching formula for $q$-Littlewood-Richardson coefficients'' in Formal Power Series and Algebraic Combinatorics (Scottsdale, Ariz., 2001), Adv. in Appl. Math. 30, Elsevier, San Diego, Calif., 2003, 258--272.

Mathematical Reviews (MathSciNet): MR1979794

Digital Object Identifier: doi:10.1016/S0196-8858(02)00535-3

M.-P. Schützenberger, ``La correspondance de Robinson'' in Combinatoire et représentation du groupe symétrique (Strasbourg, 1976), Lecture Notes in Math. 579, Springer, Berlin, 1977, 59--113.

Mathematical Reviews (MathSciNet): MR0498826

R. P. Stanley, Invariants of finite groups and their applications to combinatorics, Bull. Amer. Math. Soc. (N.S.) 1 (1979), 475--511.

Mathematical Reviews (MathSciNet): MR0526968

Digital Object Identifier: doi:10.1090/S0273-0979-1979-14597-X

Project Euclid: euclid.bams/1183544328

--------, Enumerative Combinatorics, Vol. 2, Cambridge Stud. Adv. Math. 62, Cambridge Univ. Press, Cambridge, 1999.

Mathematical Reviews (MathSciNet): MR1676282

D. W. Stanton and D. E. White, A Schensted algorithm for rim hook tableaux, J. Combin. Theory Ser. A 40 (1985), 211--247.

Mathematical Reviews (MathSciNet): MR0814412

Digital Object Identifier: doi:10.1016/0097-3165(85)90088-3

previous :: next