Macdonald polynomials are indeed related to constant term identities. Therefore we get a new proof of the existence theorem. The form of these generic polynomials is that of a bethe eigenfunction and they imitate, on a more elementary level, the r. The ramyip formula 25 is a monomial formula for macdonald polynomials of arbitrary type, which is expressed in terms of combinatorial objects called alcove walks.
In 5 we compare the new formula to our earlier one for the symmetric case. A combinatorial formula for macdonald polynomials arun rama,b, martha yipb. A combinatorial formula for the macdonald polynomials was recently proved by haglund, haiman, and loehr 5. This result can be seen as a special case of the more general combinatorial formula for macdonald polynomials. Loehr hhl08 found a combinatorial formula for computing e. In mathematics, an llt polynomial is one of a family of symmetric functions introduced by alain lascoux, bernard leclerc, and jeanyves thibon 1997 as qanalogues of products of schur functions j. Ian grojnowski and mark haiman preprint proved a positivity conjecture for llt.
The more important description of theschurfunctions is combinatorial in nature. To our knowledge, a suitable definition of an integral form of the macdonald polynomials is still unknown in the general root system setting. 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 polynomials abstract. The main points in the theory are duality theorems 4. Two q,tsymmetry problems in symmetric function theory open. 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. A new recursion for threecolumn combinatorial macdonald polynomials elizabeth niese dept. Macdonald polynomials, and explain the knopsahi recurrence.
A combinatorial formula for the modified macdonald polynomials was proved in. A combinatorial model for the macdonald polynomials pnas. The canonical reference on modified macdonald polynomials is the book by jim haglund, hag07. Combinatorial formula for macdonald polynomials, bethe ansatz, and generic macdonald polynomials olshanski, shifted jack polynomials, binomial formula, and applications, math. The main application of schur polynomials in representation theory of the. Jacobi gave the definition of schur polynomials via cauchys bialternant formula and proved the jacobitrudi identity.
It is also possible to get a formula as sum over ssyts and the cocharge statistic. Remmel chair chang, carol haekyung, geometric interpretations of the macdonald polynomials and the n. In fact, that formula can be recast as a sum of certain f tz. 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. Motivationsymmetric functionsmacdonald polynomialsmacdonald interpolation polynomials macdonald polynomials made easy s. A recursion and a combinatorial formula for jack polynomials. Heckealgebrarepresentation was written by anne schilling and. We prove that this specialization expands nonnegatively into the fundamental slide polynomials, introduced by the author and searles.
It is also possible to get a combinatorial description using semistandard augmented fillings, by specializing the combinatorial formula for macdonald polynomials see wikipedia on the macdonald polynomials. Definition and formulas for modified 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. 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. Cauchy in, where he defined the schur functions as a ratio of alternants, and proving that these are symmetric. Publications and preprints sami assaf usc dana and david. These formulas resemble the formulas of haglund, haiman and loehr for macdonald polynomials of type gl n.
Recently, ram and yip gave a formula for the macdonald polynomials of arbitrary type in terms of the corresponding af. Dual equivalence graphs, ribbon tableaux and macdonald. In fact, that formula can be recast as a sum of certain f t z. Loehr 2005 showed how to expand macdonald polynomials in terms of llt polynomials. This work records the details of the ramyip formula for nonsymmetric macdonaldkoornwinder polynomials for the double affine hecke algebras of notnecessarilyreduced affine root systems.
Return the nonsymmetric macdonald polynomial in type a corresponding to a shape mu, with basement permuted according to pi. We prove the new formula by verifying that it satisfies a recurrence, due to knop, that characterizes the nonsymmetric macdonald polynomials. Combinatorial definition of halllittlewood polynomials sum. It is shown that the t0 equalparameter specialization of nonsymmetric macdonald polynomials admits an explicit combinatorial formula in terms of quantum alcove paths, generalizing the formula of lenart in. Combinatorial formula for macdonald polynomials, bethe ansatz. A combinatorial formula for macdonald polynomials arxiv. Combinatorial definition of halllittlewood polynomials. Carini, luisa, combinatorial methods for computing plethysms of schur functions 1995, jeffrey b.
C normalization edit the jack functions form an orthogonal basis in a space of symmetric polynomials, with inner product. Galleries, halllittlewood polynomials, and structure constants of the spherical hecke algebra. Transformed macdonald symmetric functions for many reasons that we shall not discuss here, the following transformed macdonald polynomials play a fundamental role in the theory. A combinatorial model for the macdonald polynomials. He also posed the problem of finding a combinatorial rule to describe these polynomials. This gives a uniform formula for both hook and two column partitions.
Macdonald originally associated his polynomials with weights. Our methods also provide a definition and positivity theorem for llt polynomials of other types. He later introduced a nonsymmetric generalization in 1995. Information and translations of macdonald polynomials in the most comprehensive dictionary definitions resource on the web. These formulas are generalizations of the formulas of haglundhaimanloehr for macdonald polynoimals of type gln. On combinatorial formulas for macdonald polynomials. Finally, we give a table of type a nonsymmetric macdonald polynomials in an appendix, for the. In this paper we use the combinatorics of alcove walks to give uniform combinatorial formulas for macdonald polynomials for all lie types.
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. Conjecture 1998, mark haiman chair chen, youngming, combinatorial algorithms for plethysm 1982, adriano m. Jan 01, 2009 a combinatorial formula for macdonald polynomials. Morse, recursions and explicit formulas for particular nvariable sahiknop and macdonald polynomials. Definition of macdonald polynomials in the dictionary. While very useful for computation and interesting in its own right, this combinatorial formula does not immediately imply positivity of the kostka macdonald coefficients, as it gives the decomposition of the macdonald polynomials into monomial symmetric functions rather than into schur functions.
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. The form of these generic polynomials is that of a bethe eigenfunction and they imitate, on a more elementary. Ram and yip gave a formula for the macdonald polynomials of arbitrary type in terms of socalled alcove walks. At q0 these formulas specialize to the formula of schwer for the macdonald spherical function in terms of positively folded. An analogue of the robinsonschenstedknuth algorithm and. Macdonald introduced a remarkable new basis for the space of symmetric functions.
Note that if both \q\ and \t\ are specified, then they must have the same parent. Combinatorial formulas for macdonald and halllittlewood. A combinatorial model for the macdonald polynomials europe. Jan 12, 2020 due to the definition of the ideals, the macdonald polynomials exhibit conjugate symmetry in and in the sense that. The canonical reference on modified macdonald polynomials is the book by jim haglund. The theory of nonsymmetric macdonald polynomials was developed by cherednik, macdonald and opdam. Our main tool is a qintegral representation for ordinary macdonald polynomial. Definition and formulas for macdonald e polynomials specialized. Ian grojnowski and mark haiman preprint proved a positivity conjecture for llt polynomials that combined with the previous result implies the macdonald positivity conjecture for macdonald polynomials, and extended the definition of. Give a combinatorial proof of the conjugate symmetry of the summation in equation. A combinatorial formula for macdonald polynomials sciencedirect. A combinatorial formula for nonsymmetric macdonald.
Sahi and give two explicit formulas for these polynomials. Our result, combined with the combinatorial formula of haglund, haiman and loehr for macdonald polynomials, yields a new proof of the positivity theorem for macdonald polynomials. Haglund recently proposed a combinatorial interpretation of the modified macdonald polynomials h. Prove bonnets recursion formula for legendre polynomials. 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. 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. We study the specialization of the type a nonsymmetric macdonald polynomials at \t0\ based on the combinatorial formula of haglund, haiman, and loehr. Combinatorial formula for macdonald polynomials, bethe. From their definition, all one can infer is that the k. A combinatorial formula for macdonald polynomials request pdf. Affine hecke algebras and positivity of llt and macdonald. Ole warnaar department of mathematics and statistics.
The first definition of nonsymmetric macdonald polynomials is quite cumbersome and indirect. 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. In 3 we state our main theorem, the combinatorial formula. 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. A combinatorial formula for the modified macdonald polynomials was proved in, where a close connection with llt polynomials is made apparent.
We give a combinatorial proof of this conjecture, which establishes the existence and integrality of h. The mathematical field of combinatorics involves determining the number of possible choices for a subset. In this paper we use the combinatorics of alcove walks to give a uniform combinatorial formula for macdonald polynomials for all lie. 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. Later, an alternative combinatorial formula using alcove walks was proved by a. Combinatorial formula for macdonald polynomials and generic. Publications and preprints sami assaf usc dana and. A new recursion for threecolumn combinatorial macdonald. 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. 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. Specializations of nonsymmetric macdonaldkoornwinder.
Moreover, all the binomials are assumed to be expanded in the second variable. Cherednik algebras, macdonald polynomi als and combinatorics. Combinatorial formula for the macdonald polynomialsedit. In this lesson, we use examples to explore the formulas that describe four combinatoric. Finally, we give a table of type a nonsymmetric macdonald polynomials in. We further introduce a general family of polynomials f t z. Building on this work, haglund, haiman, and loehr 4 derive a combinatorial formula for nonsymmetric macdonald polynomials, which gives a new decomposition of the macdonald polynomial into nonsymmetric components. Nonsymmetric macdonald polynomials sage reference manual. Morse, sahiknop and macdonald polynomials related to qultraspherical functions and basic hypergeometric series. We extend some results about shifted schur functions to the general context of shifted macdonald polynomials. As a corollary, this yields a combinatorial formula for the kostka macdonald polynomials for partitions with at most 3 columns. In this paper we use the combinatorics of alcove walks to give a uniform combinatorial formula for macdonald polynomials for all lie types. The formula for macdonald polynomials which we give in theorem 3.
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