The jack functions form an orthogonal basis in a space of symmetric polynomials, with inner product. Dual equivalence graphs, ribbon tableaux and macdonald. Department of mathematics, university of pennsylvania, philadelphia, pa 19104 6395. Various works 19,14,10,3,15 establish several combinatorial properties of these polynomials and conjecture others. Macdonald polynomials are indeed related to constant term identities.
We prove the new formula by verifying that it satisfies a recurrence, due to knop and sahi, that characterizes the nonsymmetric macdonald polynomials. We prove the new formula by verifying that it satisfies a recurrence, due to knop, that characterizes the 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. Morse, sahiknop and macdonald polynomials related to qultraspherical functions and basic hypergeometric series. Ole warnaar department of mathematics and statistics. In this paper we use the combinatorics of alcove walks to give a uniform combinatorial formula for macdonald polynomials for all lie types. 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 prove a combinatorial formula for the macdonald polynomial. For a partition let mi be the number of parts which are equal to i and let u.
More recently, combinatorial formulas for both the macdonald polynomials and for re n have been discovered. We prove the new formula by verifying that it satis. Polynomial division is equivalent to deconvolution. He also posed the problem of finding a combinatorial rule to describe these polynomials. We give a direct proof of the combinatorial formula for interpolation macdonald polynomials by introducing certain polynomials, which we call generic macdonald. Pursuant work with kschur functions has led to a theory that branches into many fields along the lines of modern schubert calculus. Note that this is a different reading order than the usual french or japanese ones. Citeseerx a combinatorial formula for macdonald polynomials. This celebrated combinatorial formula brought a breakthrough in macdonald polynomial theory.
Citeseerx document details isaac councill, lee giles, pradeep teregowda. In a previous paper 6, we gave a combinatorial formula. In this paper we use the combinatorics of alcove walks to give a uniform combinatorial formula for macdonald polynomials for all lie. Dual equivalence graphs, ribbon tableaux and macdonald polynomials by sami hayes assaf doctor of philosophy in mathematics university of california, berkeley professor mark haiman, chair we make a systematic study of a new combinatorial construction called a dual equivalence graph.
We derive an explicit sum formula for symmetric macdonald polynomials. The form of these generic polynomials is that of a bethe eigenfunction and they imitate, on a more elementary level, the. A recursion and a combinatorial formula for jack polynomials. These formulas are sums over llings of tableaulike diagrams. 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. It is also possible to compute the littlewoodrichardson coe cients for such polynomials see 16,17, but currently there are no combinatorial formulas. 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 summation formula for macdonald polynomials springerlink. The kschur functions are symmetric functions discovered in a study of macdonald polynomials. General formula for polynomial division mathematics. Nonsymmetric macdonald polynomials sage reference manual.
These formulas are generalizations of the formulas of haglundhaimanloehr for macdonald polynoimals of type gln. In this paper, we prove that the integral form of macdonald polynomials j. A combinatorial formula for nonsymmetric macdonald. From their definition, all one can infer is that the k. Macdonald originally associated his polynomials with weights. Haglund proceedings of the national academy of sciences nov 2004, 101 46 16127161. A combinatorial formula for macdonald polynomials ams journal. A combinatorial model for the macdonald polynomials ncbi.
Galleries, halllittlewood polynomials, and structure constants of the spherical hecke algebra. A combinatorial formula for nonsymmetric macdonald polynomials. The more important description of theschurfunctions is combinatorial in nature. 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. Morse, recursions and explicit formulas for particular nvariable sahiknop and macdonald polynomials. Department of mathematics, the college of william and mary, williams burg, va. Introduction in a previous paper 6, we gave a combinatorial formula for macdonald s symmetric functions for root systems of type a. A new recursion for threecolumn combinatorial macdonald polynomials elizabeth niese dept. The form of these generic polynomials is that of a bethe eigenfunction and they imitate, on a more. A combinatorial model for the macdonald polynomials j. Combinatorial formulas for macdonald and halllittlewood. A combinatorial formula for macdonald polynomials citeseerx.
On combinatorial formulas for macdonald polynomials. A combinatorial formula for macdonald polynomials sciencedirect. Heckealgebrarepresentation was written by anne schilling and. Of recent note in the theory of macdonald polynomials has been the success of haglundhaimanloehr in giving, in the type gl n case, explicit combinatorial formulas for the expansion of macdonald polynomials in terms of monomials. Combinatorial formula for macdonald polynomials, bethe.
Return the nonsymmetric macdonald polynomial in type a corresponding to a shape mu, with basement permuted according to pi. This is a corollary of our realization of kr crystals in terms of the quantum alcove model and the ramyip formula for macdonald polynomials 44. 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 terms of fillings of young diagrams. An enormous amount of combinatorics, group theory, and algebraic geometry is coded into these polynomials. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. From macdonald polynomials to a charge statistic beyond type a, journal of combinatorial theory series a, v. Motivationsymmetric functionsmacdonald polynomialsmacdonald interpolation polynomials macdonald polynomials made easy s. Combinatorial formula for macdonald polynomials, bethe ansatz, and generic macdonald polynomials. 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.
More importantly, our formulas are explicit enough such that both the recursion relation and the combinatorial formula enable us to prove a conjecture of macdonald and stanley m, s. We give a combinatorial proof of this conjecture, which establishes the existence and integrality of h. A combinatorial model for the macdonald polynomials. Combinatorial formula for macdonald polynomials and. A combinatorial formula for macdonald polynomials arxiv. 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. He later introduced a nonsymmetric generalization in 1995. 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. Corollaries to our main theorem include the expansion of.
In this paper we use the combinatorics of alcove walks to give uniform combinatorial formulas for macdonald polynomials for all lie types. A combinatorial formula for macdonald polynomials 3 here and throughout. Note that if both \q\ and \t\ are specified, then they must have the same parent. This result can be seen as a special case of the more general combinatorial formula for macdonald polynomials. A combinatorial model for the macdonald polynomials pnas. Ram and yip gave a formula for the macdonald polynomials of arbitrary type in terms of socalled alcove walks. At q 0 these formulas specialize to the formula of schwer for. On combinatorial formulas for macdonald polynomials nasaads. These formulas resemble the formulas of haglund, haiman and loehr for macdonald polynomials of type gl n. In fact, we gave two formulas, one for the transformed macdonald polynomials hx\ q, t. Unfortunately, it does not give any combinatorial description of k. In 19, haiman, haglund, loehr, remmel, and ulyanov put forth the \shu e conjecture, which states that the frobenius characteristic of the ring r n has the combinatorial formula re n x p2pfn qareaptdinv px where pf. Haglund and proved by haglundhaimanloehr in hhl1 and hhl2.
A probabilistic interpretation of the macdonald polynomials. Crystal energy functions via the charge in types a and c, mathematische zeitschrift, v. Our expression contains multiple sums over the symmetric group and uses the action of hecke generators on the ring of polynomials. Haglund recently proposed a combinatorial interpretation of the modified macdonald polynomials h. Haglund and proved by haglund, haiman and loehr in 4 and 5. The partitions of a given n are partially ordered by 3. Our proof proceeds by constructing constructing combinatorial formula for the schur coefficients when.
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