In representation theory they are the characters of polynomial irreducible representations of the general linear groups. Macdonald polynomials and geometry mark haiman abstract. We construct a combinatorial formula for the hilbert series of garsiahaiman modules as a sum over standard young tableaux, and provide a bijection between a group of fillings and the corresponding standard young tableau in the hook shape case. Symmetry of the specialized macdonald polynomials 38 4. In this paper, we use the combinatorics of alcove walks to give a uniform combinatorial formula for macdonald polynomials for all lie types. Introduction in a previous paper 6, we gave a combinatorial formula for macdonalds symmetric functions for root systems of type a. A combinatorial formula for the macdonald polynomials is proved by haglund, haiman, and loehr in 2. On combinatorial formulas for macdonald polynomials. The form of these generic polynomials is that of a bethe eigenfunction and they imitate, on a more elementary. Jan 01, 2009 a combinatorial formula for macdonald polynomials. Mar 15, 2005 it is clear that we have here a new powerful tool of proof which many of us researchers in macdonald polynomial theory are anxious to apply to the resolution of the wide variety of conjectures that are still unproven, the top of the list being the 20yearold search for a combinatorial interpretation for the macdonald q, tkostska polynomials k. Nov 16, 2004 a combinatorial model for the macdonald polynomials j. We study bigraded snmodules introduced by garsia and haiman as an approach to prove the macdonald positivity conjecture.
This celebrated combinatorial formula brought a breakthrough in macdonald polynomial theory. The immediate purpose of this paper is to establish a combinatorial formula for the schur expansion of the kribbon tableaux generating functions known as llt polynomials when k 3. It is also possible to get a formula as sum over ssyts and the cocharge statistic. A combinatorial formula for macdonald polynomials article pdf available in journal of the american mathematical society 183 september 2004 with 39 reads how we measure reads. In this paper we use the combinatorics of alcove walks to give a uniform combinatorial formula for macdonald polynomials for all lie types. Haglundhaimanloehr hhl combinatorial formula, namely the permutedbasement macdonald polynomials, see 1,9. A new recursion for threecolumn combinatorial macdonald polynomials elizabeth niese dept. As a corollary, this yields a combinatorial formula for the kostkamacdonald polynomials for partitions with at most 3 columns.
Cherednik algebras, macdonald polynomials and combinatorics 845 of x is a re. We give an explicit combinatorial formula for the schur expansion of macdonald polynomials indexed by partitions with second part at most two. A combinatorial formula for nonsymmetric macdonald. In this paper we use the combinatorics of alcove walks to give uniform combinatorial formulas for macdonald polynomials for all lie types. Corollaries to our main theorem include the expansion of in terms of llt polynomials, a new proof of the charge formula of lascoux and schutzenberger for halllittlewood polynomials, a new proof of knop and sahis combinatorial formula for jack polynomials as well as a lifting of their formula to integral form macdonald polynomials, and a new. We give a direct proof of the combinatorial formula for interpolation macdonald polynomials by introducing certain polynomials, which we call generic macdonald polynomials, and which depend on d additional parameters and specialize to all macdonald polynomials of degree d. These formulas are generalizations of the formulas of haglundhaimanloehr for macdonald polynoimals of type gln. This method is based on macdonald formula for halllittlewood polynomial and extends the result of. A combinatorial formula for the modified macdonald polynomials was proved in, where a close connection with llt polynomials is made apparent. A combinatorial formula for macdonald polynomials ams journal. Pursuant work with kschur functions has led to a theory that branches into many fields along the lines of modern schubert calculus.
Properties of nonsymmetric macdonald polynomials at q. Haglund and proved by haglund, haiman and loehr in 4 and 5. Positivity for special cases of,kostka coefficients and standard tableaux. Note that this is a different reading order than the usual french or japanese ones. Combinatorial definition of halllittlewood polynomials. It is also possible to get a combinatorial description using semi standard augmented fillings, by specializing the combinatorial formula for macdonald polynomials see wikipedia on the macdonald polynomials. A combinatorial formula for fusion coefficients, dmtcs. Feb 22, 2005 as corollaries, we obtain the cocharge formula of lascoux and schutzenberger for halllittlewood polynomials, a formula of sahi and knop for jacks symmetric functions, a generalization of this result to the integral macdonald polynomials j. Cherednik algebras, macdonald polynomi als and combinatorics. Combinatorial formulas introduction the macdonald polynomials were introduced in the mid 1980s 11,12 as a remarkable family orthogonal polynomials generalizing the spherical functions for a padic group, the weyl aracters, the jack. Unfortunately, it does not give any combinatorial description of k. A combinatorial model for the macdonald polynomials pnas. The proof comes as a corollary to the result that generalized dual equivalence classes of permutations are in explicit bijection with unions of standard dual equivalence.
In this lesson, we use examples to explore the formulas that describe four combinatoric. Ram and yip gave a formula for the macdonald polynomials of arbitrary type in terms of socalled alcove walks. Variants of the rsk algorithm adapted to combinatorial. Haglunds combinatorial formula for macdonald polynomials ian macdonald introduced the symmetric polynomials now called macdonald polynomials in 1988 19. Recently, an alcove walk model was given for these, as well, see 7,8. In this paper we use the combinatorics of alcove walks to give a uniform combinatorial formula for macdonald polynomials for all lie. The main points in the theory are duality theorems 4. Loehr hhl08 found a combinatorial formula for computing e. From haglunds formula for the transformed macdonald polynomials, this has the further consequence of a combinatorial expansion of the macdonald kostka polynomials indexed by a partition with at most \3\ columns. We conjecture that our polynomials are in fact equal to the skew macdonald polynomials. This generalizes the alcove walk model by ram and yip 24 for general type nonsymmetric macdonald polynomials. A combinatorial formula for the modified macdonald polynomials was proved in. Combinatorial formula for macdonald polynomials and generic.
These formulas are generalizations of the formulas of haglundhaimanloehr for macdonald polynomials of type gln. This implies the schur positivity of the kribbon tableaux generating functions introduced by lascoux, leclerc and thibon. In fact, that formula can be recast as a sum of certain f tz. The form of these generic polynomials is that of a bethe eigenfunction and they imitate, on a more elementary level, the r. Combinatorial formulas connected to diagonal harmonics. Combinatorial formula for macdonald polynomials, bethe. Crystal energy functions via the charge in types a and c, mathematische zeitschrift, v. A combinatorial formula for macdonald polynomials arxiv. This method is based on the combinatorial formula of haglund, haiman and loehr for the macdonald polynomials and extends the result of a. We give a combinatorial proof of this conjecture, which establishes the existence and integrality of h.
Combinatorial formulas for macdonald and halllittlewood polynomials cristian lenart department of mathematics and statistics, state university of new york at albany, albany, ny 12222 a breakthrough in the theory of type a macdonald polynomials is due to haglund, haiman and loehr, who exhibited a combinatorial formula for these polynomials in. In a previous paper 6, we gave a combinatorial formula for macdonalds. The s, which are now called macdonald polynomials, specialize to many of the well known bases for the symmetric functions, by suitable choices of the parameters q and t. The proof comes as a corollary to the result that generalized dual equivalence classes of permutations are unions of standard dual equivalence classes of permutations for certain. This method is based on macdonald formula for halllittlewood polynomial and extends the result of a. Haglund proceedings of the national academy of sciences nov 2004, 101 46 16127161. A combinatorial formula for nonsymmetric macdonald polynomials. If a is a polynomial or formal series, pka denotes the result of substituting ak for each indeterminate a appearing in a including q and t. At q 0 these formulas specialize to the formula of schwer for the macdonald spherical function in terms of positively folded alcove walks and at q t. We give a direct proof of the combinatorial formula for interpolation macdonald polynomials by introducing certain polynomials, which we call generic macdonald polynomials, which depend on d additional parameters and specialize to all macdonald polynomials of degree d. Publications and preprints sami assaf usc dana and. We prove a combinatorial formula for the macdonald polynomial. Macdonald originally associated his polynomials with weights.
The first definition of nonsymmetric macdonald polynomials is quite cumbersome and indirect. Of recent note in the theory of macdonald polynomials has been the success of haglund, haiman and loehr in giving, in the type gln case, explicit combinatorial formulas for the expansion of macdonald polynomials in terms of monomials. Macdonald introduced them in 5 and conjectured that their expansion in terms of schur polynomials should have positive coe cients. He later introduced a nonsymmetric generalization in 1995. A combinatorial formula for macdonald polynomials sciencedirect. The paper 35 gave uniform formulas for the expansion of macdonald polynomials of all lie types, and this was achieved by using the alcove walk model as a tool for expanding. A combinatorial model for the macdonald polynomials. Citeseerx combinatorial formula for the hilbert series of.
Haglund recently proposed a combinatorial interpretation of the modified macdonald polynomials h. Introduction the immediate purpose of this paper is to establish a combinatorial formula for the schur coe. A combinatorial formula for macdonald polynomials arun rama,b, martha yipb. A new recursion for threecolumn combinatorial macdonald. The mathematical field of combinatorics involves determining the number of possible choices for a subset. Birs workshop on applications of macdonald polynomials. Combinatorial formula for macdonald polynomials and. A corresponding combinatorial formula for the integral form macdonald polynomials, j. Citeseerx a combinatorial formula for the hilbert series. In mathematics, schur polynomials, named after issai schur, are certain symmetric polynomials in n variables, indexed by partitions, that generalize the elementary symmetric polynomials and the complete homogeneous symmetric polynomials. These formulas are sums over llings of tableaulike diagrams. Prove bonnets recursion formula for legendre polynomials. A recent breakthrough in the theory of type a macdonald polynomials is due to haglund, haiman and loehr, who exhibited a combinatorial formula for these polynomials in terms of a pair of statistics on fillings of young diagrams. Publications and preprints sami assaf usc dana and david.
Toward the schur expansion of macdonald polynomials nasaads. These conditions arise by setting q t 0 in the combinatorial formula for the nonsymmetric macdonald polynomials 4. Breakthroughs in the theory of macdonald polynomials. Combinatorial formula for macdonald polynomials, bethe ansatz. Cherednik algebras, macdonald polynomials and combinatorics. We prove the new formula by verifying that it satisfies a recurrence, due to knop, that characterizes the nonsymmetric macdonald polynomials. Properties of nonsymmetric macdonald polynomials at q 1and q. Moreover, we construct an association of the fillings giving the monomial terms of macdonald polynomials with the standard young tableaux. Recent advances in the combinatorial theory of macdonald polynomials include the work by haglund, haiman and loehr 14 and 15 who, in the type acase, gave explicit combinatorial formulas for the expansion of macdonald polynomials in terms of monomials. The canonical reference on modified macdonald polynomials is the book by jim haglund, hag07. We explain some remarkable connections between the twoparameter symmetric polynomials discovered in 1988 by macdonald, and the geometry of certain algebraic varieties, notably the hilbert scheme hilbnc 2 of points in the plane, and the varietycnof pairs of commuting n nmatrices. These formulas resemble the formulas of haglund, haiman and loehr for macdonald polynomials of type gl n. Macdonald introduced a remarkable new basis for the space of symmetric functions.
Combinatorial formulas connected to diagonal harmonics and. I n 1988, kadell 1 conjectured the existence of a family of symmetric polynomials, depending on. Citeseerx a combinatorial formula for the hilbert series of. Pdf a combinatorial formula for macdonald polynomials.
Sahis combinatorial formula for jack polynomials as well as a lifting of their formula to integral form macdonald polynomials, and a new combinatorial rule for the kostkamacdonald coe. Dual equivalence graphs, ribbon tableaux and macdonald. We only consider the type a for which there is a combinatorial rule, discovered by haglund et al. As corollaries, we obtain the cocharge formula of lascoux and schutzenberger for halllittlewood polynomials, a formula of sahi and knop for jacks symmetric functions, a. There are several versions of macdonald polynomials, each of which can be defined algebraically as the unique symmetric polynomials satisfying certain orthogonality or triangularity conditions.
We introduce a combinatorial way of calculating the hilbert series of bigraded snmodules as a weighted sum over standard young tableaux in the hook shape case. Definition and formulas for modified macdonald polynomials. Galleries, halllittlewood polynomials, and structure constants of the spherical hecke algebra. The kschur functions are symmetric functions discovered in a study of macdonald polynomials. Citeseerx combinatorial formula for the hilbert series. Combinatorial definition of halllittlewood polynomials sum. We prove the new formula by verifying that it satis.
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