A probabilistic interpretation of the macdonald polynomials. The shu\u000fe conjecture gives an elegant expression of the frobenius series of the diagonal harmonics. Sep 01, 2003 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. We describe two of them here, both of which ask to explain the symmetry between and exhibited by certain combinatorially defined polynomials in. A new recursion for threecolumn combinatorial macdonald polynomials elizabeth niese dept. Our formula does, however, reduce the problem to a special case of the conjecture in 20 that llt polynomials have positive expansions in terms of schur polynomials. Combinatorial formula for macdonald polynomials and generic. A combinatorial formula for macdonald polynomials arun rama,b, martha yipb. Combinatorial formula for macdonald polynomials and. This celebrated combinatorial formula brought a breakthrough in macdonald polynomial theory. A combinatorial formula for the associated legendre.
This gives a uniform formula for both hook and two column partitions. About 170 170 1 7 0 years ago, a young mathematician by the name of henrik abel proved that it is impossible to find a formula for the solutions of a quintic polynomial by adding, subtracting, multiplying, dividing and taking n th n\textth n th roots. Haglund recently proposed a combinatorial interpretation of the modified macdonald polynomials h. A combinatorial model for the macdonald polynomials pnas. Citeseerx document details isaac councill, lee giles, pradeep teregowda.
We derive an explicit sum formula for symmetric macdonald polynomials. Combinatorics of macdonald polynomials and cyclic sieving 7 llings that is, nonattacking llings with no coinversions and the specialization. Pdf a combinatorial formula for macdonald polynomials. Combinatorial formula for macdonald polynomials, bethe. A nonsymmetric version of okounkovs bctype interpolation. In fact, that formula can be recast as a sum of certain f tz. 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. Ram and yip gave a formula for the macdonald polynomials of arbitrary type in terms of socalled alcove walks. The shu e conjecture gives an elegant expression of the frobenius series of the diagonal harmonics. We study the specialization of the type a nonsymmetric macdonald polynomials at \t0\ based on the combinatorial formula of haglund, haiman, and loehr. 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. Combinatorial and umbral methods for orthogonal polynomials. Unfortunately, it does not give any combinatorial description of k. Citeseerx combinatorial formula for macdonald polynomials.
A summation formula for macdonald polynomials springerlink. We will explore the combinatorial and geometric properties related to the macdonald polynomials and the diagonal harmonics. That conjecture is known to hold for llt polynomials indexed by tuples of partition diagrams 9, 24. Combinatorics of macdonald polynomials and cyclic sieving joakim uhlin. 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.
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 mathematical field of combinatorics involves determining the number of possible choices for a subset. Haglund proceedings of the national academy of sciences nov 2004, 101 46 16127161. Recently, ram and yip gave a formula for the macdonald polynomials of arbitrary type in terms of the corresponding af. Combinatorial formula for macdonald polynomials, bethe ansatz, and generic macdonald polynomials. 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 and umbral methods for orthogonal polynomials a dissertation presented to the faculty of the graduate school of arts and sciences of brandeis university, waltham, massachusetts by pallavi jayawant this thesis gives generating functions for various classical orthogonal polynomials. 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.
Our proof proceeds by constructing constructing combinatorial formula for the schur coefficients when. Binomial formula for macdonald polynomials and its applications andrei okounkov abstract. In this paper we use the combinatorics of alcove walks to give a uniform combinatorial formula for macdonald polynomials for all lie. A combinatorial formula for the associated legendre functions of integer degree j. Two q,tsymmetry problems in symmetric function theory. 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. Cherednik algebras, macdonald polynomi als and combinatorics. Fall 2005spring 2006 combinatorics seminar schedule.
This is a symmetric function because the numerator and denominator are both alternating, and a polynomial since all alternating polynomials are divisible by the vandermonde determinant. He also posed the problem of finding a combinatorial rule to describe these polynomials. Nov 16, 2004 a combinatorial model for the macdonald polynomials j. Combinatorial formulas for macdonald and halllittlewood. 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. Haglunds formula opened the way to new explorations of the combinatorial properties of macdonald polynomials and the associated hilbert series fe. Publications and preprints sami assaf usc dana and. 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. 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. A new recursion for threecolumn combinatorial macdonald. The form of these generic polynomials is that of a bethe eigenfunction and they imitate, on a more elementary level, the r. While the geometric properties of the hilbert scheme and schemes.
In 2007, haglund, haiman and loehr gave a combinatorial formula for the nonsymmetric macdonald polynomials. An overview of the geometry and combinatorics of the. Combinatorial aspects of macdonald and related polynomials. Im kind of sad to seemingly be stuck with weird combinatorial arguments again. Haglund and proved by haglund, haiman and loehr in 4 and 5. Combinatorial formula for macdonald polynomials, bethe ansatz. We extend some results about shifted schur functions to the general context of shifted macdonald polynomials. Two q,tsymmetry problems in symmetric function theory open. There are many natural combinatorial problems yet to be solved in the study of twoparameter symmetric functions such as macdonald polynomials. The set is the set of all fillings of the young diagram of shape in which each square is filled with a positive integer with no other restrictions on the entries.
In fact, our formula yields a lift of the knopsahi formula from jack polynomials j. 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. Sahi and give two explicit formulas for these polynomials. We give an explicit combinatorial formula for the schur expansion of macdonald polynomials indexed by partitions with second part at most two.
Combinatorics of macdonald polynomials and cyclic sieving 5 contents abstract 2 sammanfattning 3 acknowledgement 4 1. A combinatorial model for the macdonald polynomials. As corollaries, we obtain the cocharge formula of lascoux and. A combinatorial formula for macdonald polynomials request pdf. Jan 12, 2020 the frobenius characteristic of is the transformed macdonald polynomial, which was shown in to exhibit the following combinatorial formula. An enormous amount of combinatorics, group theory, and algebraic geometry is coded into. Publications and preprints sami assaf usc dana and david. Our main tool is a qintegral representation for ordinary macdonald polynomial. 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. From their definition, all one can infer is that the k. We give a combinatorial proof of this conjecture, which establishes the existence and integrality of h.
Combinatorial formula for macdonald polynomials, bethe ansatz, and generic macdonald polynomials by andrei okounkov get pdf 181 kb. A nonsymmetric version of okounkovs bctype interpolation macdonald polynomials tom koornwinder kortewegde vries institute, university of amsterdam. Recursions and divisibility properties for combinatorial. 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. We conjecture that our polynomials are in fact equal to the skew macdonald polynomials. The main points in the theory are duality theorems 4. A recursion and a combinatorial formula for jack polynomials.
A problem of ongoing interest involves the search for recursions characterizing the polynomials fe, as ranges over all integer partitions. Prove bonnets recursion formula for legendre polynomials. The connection with ordinary macdonald polynomials will be explained below. We prove that this specialization expands nonnegatively into the fundamental slide polynomials, introduced by the author and searles. A combinatorial formula for nonsymmetric macdonald polynomials. We generalize the binomial formula for jack polynomials proved in oo2 and consider some applications.
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. The form of these generic polynomials is that of a bethe eigenfunction and they imitate, on a more elementary. We consider homogeneous polynomials of degree n in the variables z1. A combinatorial formula for macdonald polynomials sciencedirect. 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. 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. A combinatorial formula for nonsymmetric macdonald.
He also posed the problem of finding a combinatorial rule to. In this paper we use the combinatorics of alcove walks to give uniform combinatorial formulas for macdonald polynomials for all lie types. These formulas resemble the formulas of haglund, haiman and loehr for macdonald polynomials of type gl n. Combinatorics of macdonald polynomials and cyclic sieving. In this paper, we prove that the integral form of macdonald polynomials j. In this lesson, we use examples to explore the formulas that describe four combinatoric. Our expression contains multiple sums over the symmetric group and uses the action of hecke generators on the ring of polynomials. The degree d schur polynomials in n variables are a linear basis for the space of homogeneous degree d symmetric polynomials in n variables. A combinatorial formula for macdonald polynomials arxiv. For a partition let mi be the number of parts which are equal to i. The nonsymmetric macdonald polynomials specialize to demazure characters by taking qt0, and to key polynomials when qt. The formula generalizes our previous combinatorial interpretation. The theory of nonsymmetric macdonald polynomials was developed by cherednik, macdonald and opdam.
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