A combinatorial formula for nonsymmetric macdonald. Abstract combinatorial formulas connected to diagonal harmonics and macdonald polynomials meesue yoo james haglund, advisor we study bigraded s. Polynomial factoring calculator this online calculator writes a polynomial, with one or more variables, as a product of linear factors. In this paper we use the combinatorics of alcove walks to give a uniform combinatorial formula for macdonald polynomials for all lie types. The connection with ordinary macdonald polynomials will be explained below. In elementary algebra, the binomial theorem or binomial expansion describes the algebraic expansion of powers of a binomial. On combinatorial formulas for macdonald polynomials nasaads. Since the former is given by halllittlewood polynomials, we arrive at a combinatorial expression of halllittlewood polynomials as above.
Citeseerx a combinatorial formula for macdonald polynomials. Stanley provides a pieri rule for jack polynomials. We now give three more applications of our combinatorial formulas for macdonald polynomials. Note that if both \q\ and \t\ are specified, then they must have the same parent. Citeseerx document details isaac councill, lee giles, pradeep teregowda.
Factoring formulas for cubic polynomials factoring. The form of these generic polynomials is that of a bethe eigenfunction and they imitate, on a more elementary level, the. Haglund and proved by haglund, haiman and loehr in 4 and 5. In this paper we use the combinatorics of alcove walks to give a uniform combinatorial formula for macdonald polynomials for all lie. We generalize the binomial formula for jack polynomials proved in oo2 and consider some applications. Ole warnaar department of mathematics and statistics. Haiman, tableaux formulas for macdonald polynomials, special edition in honor of christophe reutenauer 60 birthday, international journal of algebra and computation, volume 23, issue 4, 20, pp. The nonsymmetric macdonald polynomials specialize to demazure characters by taking qt0, and to key polynomials when qt. Cherednik algebras, macdonald polynomials and combinatorics 845 of x is a re. He also posed the problem of finding a combinatorial rule to describe these polynomials. Schubert calculus is the modern approach to classical problems in enumerative algebraic geometry, specifically on flag varieties and their many generalizations. This factor cancels all denominators in the hhl formula but not in ours. Recently, ram and yip gave a formula for the macdonald polynomials of arbitrary type in terms of the corresponding affine weyl group. On combinatorial formulas for macdonald polynomials core.
A nonsymmetric version of okounkovs bctype interpolation. Combinatorial formula for macdonald polynomials and. The mathematical field of combinatorics involves determining the number of possible choices for a subset. Motivationsymmetric functionsmacdonald polynomialsmacdonald interpolation polynomials macdonald polynomials made easy s. Combinatorial aspects of macdonald and related polynomials. Sahis combinatorial f or m ula for jack polynomials as well as a lifting of their formula to in te gral form macdonald p olynomia ls, a nd a new combinatorial r ule for the kostk a macdonald. A combinatorial formula for nonsymmetric macdonald polynomials. In this lesson, we use examples to explore the formulas that describe four combinatoric. We prove a combinatorial formula for the macdonald polynomial. Pdf a combinatorial formula for macdonald polynomials. Project muse a combinatorial formula for nonsymmetric. At q 0 these formulas specialize to the formula of schwer for. Combinatorial formula for macdonald polynomials, bethe.
Factorization formulas for macdonald polynomials hal. The formula for macdonald polynomials which we give in theorem 3. The form of these generic polynomials is that of a bethe eigenfunction and they imitate, on a more. In 2007, haglund, haiman and loehr gave a combinatorial formula for the nonsymmetric macdonald polynomials. On combinatorial formulas for macdonald polynomials. In a previous paper 6, we gave a combinatorial formula for macdonalds. These formulas are generalizations of the formulas of haglundhaimanloehr for macdonald polynoimals of type gln. In this thesis, we study the nonsymmetric macdonald polynomials e x. Ram and yip gave a formula for the macdonald polynomials of arbitrary type in terms of socalled alcove walks.
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. The more important description of theschurfunctions is combinatorial in nature. In fact, that formula can be recast as a sum of certain f tz. Abstract 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. We prove the new formula by verifying that it satisfies a recurrence, due to knop, that characterizes the nonsymmetric 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, and which depend on d additional parameters and specialize to all macdonald polynomials of degree d. Combinatorics of macdonald polynomials and cyclic sieving. These applications all involve the modified kostkamacdonald coefficients k. Able to display the work process and the detailed explanation. The main points in the theory are duality theorems 4.
From their definition, all one can infer is that the k. Macdonald polynomials are indeed related to constant term identities. Combinatorial formulas for macdonald and halllittlewood. We extend some results about shifted schur functions to the general context of shifted macdonald polynomials. Binomial formula for macdonald polynomials and its applications andrei okounkov abstract. Cherednik algebras, macdonald polynomials and combinatorics. A combinatorial expression of halllittlewood polynomials. Cherednik algebras, macdonald polynomi als and combinatorics. Introduction in a previous paper 6, we gave a combinatorial formula for macdonald s symmetric functions for root systems of type a. A combinatorial formula for macdonald polynomials 737 or 12, 6.
We prove the new formula by verifying that it satisfies a recurrence, due to knop and sahi, that characterizes the nonsymmetric macdonald polynomials. The formula generalizes our previous combinatorial interpretation of the integral form symmetric macdonald polynomials j. Tom koornwinder nonsymmetric okounkov interpolation polynomials. Factoring formulas factoring is nothing but breaking down a number or a polynomial into product of its factor which when multiplied together gives the original. We assign row,column coordinates to lattice squares in the first quadrant by switching the x, y coordinates of the lower left corner of the square, so the lowerleft square has coordinates 0, 0, the square above it 1, 0, etcfor a square w, we call the first coordinate of w the row value of w, denoted row. A combinatorial formula for macdonald polynomials sciencedirect. Combinatorial formula for macdonald polynomials, bethe ansatz, and generic macdonald polynomials. Recently, ram and yip gave a formula for the macdonald polynomials of arbitrary type in terms of the corresponding af. Morse, recursions and explicit formulas for particular nvariable sahiknop and macdonald polynomials. At q t 0 the symmetric macdonald polynomials are the weyl characters or schur func ons.
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