We extend some results about shifted schur functions to the general context of shifted macdonald polynomials. Citeseerx document details isaac councill, lee giles, pradeep teregowda. It is also possible to get a formula as sum over ssyts and the cocharge statistic. Just because this is a statement about the polynomials themselves rather than symmetric functions. From their definition, all one can infer is that the k. Haglund recently proposed a combinatorial interpretation of the modified macdonald polynomials h. Pdf a combinatorial formula for macdonald polynomials. In this unit we describe polynomial functions and look at some of their properties. This implies the schur positivity of the kribbon tableaux generating functions introduced by lascoux, leclerc and thibon. They are indexed by integer partitions and constitute a basis for the space of symmetric functions. Relationship between legendre polynomials and legendre functions of the second kind. We further introduce a general family of polynomials f t z. In this case our formula for the macdonald polynomial specializes to the formula for the weyl character in terms of the. Publications and preprints sami assaf usc dana and.
The jack polynomial is a homogeneous, symmetric polynomial which generalizes the schur and zonal polynomials, and is in turn generalized by the heckmanopdam polynomials and. Nov 16, 2004 he also posed the problem of finding a combinatorial rule to describe these polynomials. In our setting, the charge, an intricate combinatorial statistic appearing. On a positivity property of halllittlewood polynomials. A combinatorial expression of halllittlewood polynomials. 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. The form of these generic polynomials is that of a bethe eigenfunction and they imitate, on a more elementary level, the r.
Sahi and give two explicit formulas for these polynomials. The book is appropriate as a text for a topics course in algebraic combinatorics, a volume for selfstudy, or a reference text for researchers in any area which involves symmetric functions or lattice. A combinatorial formula for macdonald polynomials citeseerx. 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 polynomials elizabeth niese dept. The partitions of a given n are partially ordered by 3. When two polynomials are divided it is called a rational expression.
There are many natural combinatorial problems yet to be solved in the study of twoparameter symmetric functions such as macdonald polynomials. The form of these generic polynomials is that of a bethe eigenfunction and they imitate, on a more elementary. Since the former is given by halllittlewood polynomials, we arrive at a combinatorial expression of halllittlewood polynomials as above. Combinatorics of macdonald polynomials and cyclic sieving. These developments led in turn to some surprising discoveries in the combinatorics of macdonald polynomials, which are described in appendix a. From haglunds formula for the transformed macdonald polynomials, this has the further consequence of a combinatorial expansion of the macdonaldkostka polynomials indexed by a partition with at most \3\ columns. This book explains some recent applications of the theory of polynomials and algebraic geometry to combinatorics and other areas of mathematics. The formula generalizes our previous combinatorial interpretation. The jack polynomials are a family of symmetric functions which extends the schur polynomials. Macdonald originally associated his polynomials with weights. Corollaries to our main theorem include the expansion of. Combinatorial definition of halllittlewood polynomials. One of the first results in this story is a short elegant solution of the kakeya problem for finite fields, which was considered a deep and difficult problem in combinatorial geometry.
He also posed the problem of finding a combinatorial rule to describe these polynomials. Jack symmetric functions and some combinatorial properties of young symmetrizers. Binomial formula for macdonald polynomials and its applications andrei okounkov abstract. At q 0 the symmetric macdonald polynomials are the halllittlewood polynomials or the macdonald spherical functions. Jacobi gave the definition of schur polynomials via cauchys bialternant formula and proved the jacobitrudi identity. Our main tool is a qintegral representation for ordinary macdonald polynomial. We give a combinatorial proof of this conjecture, which establishes the existence and integrality of h. 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. In mathematics, the jack function is a generalization of the jack polynomial, introduced by henry jack. Before we look at the formal definition of a polynomial, lets have a look at some graphical examples. A polynomial is a mathematical expression constructed with constants and variables using the four operations.
A combinatorial formula for macdonald polynomials arxiv. A combinatorial model for the macdonald polynomials pnas. A combinatorial formula for nonsymmetric macdonald. The main application of schur polynomials in representation theory of the. These formulas resemble the formulas of haglund, haiman and loehr for macdonald polynomials of type gl n.
A combinatorial formula for macdonald polynomials sciencedirect. We generalize the binomial formula for jack polynomials proved in oo2 and consider some applications. Combinatorial formula for macdonald polynomials and generic. These formulas are sums over llings of tableaulike diagrams. A recursion and a combinatorial formula for jack polynomials. Combinatorial formula for macdonald polynomials, bethe ansatz.
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. Note that this is a different reading order than the usual french or japanese ones. In our setting, the charge, an intricate combinatorial statistic appearing in the lascouxsch. Polynomial functions and equations what is a polynomial. Two q,tsymmetry problems in symmetric function theory. Note that if both \q\ and \t\ are specified, then they must have the same parent. In other words, we have been calculating with various polynomials all along. There has been a lot of work devoted to understanding macdonald polynomials from a combinatorial point of view. Combinatorial definition of halllittlewood polynomials sum. Ram and yip gave a formula for the macdonald polynomials of arbitrary type in terms of socalled alcove walks. Cauchy in, where he defined the schur functions as a ratio of alternants, and proving that these are symmetric. Combinatorial formulas for macdonald and halllittlewood.
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. A combinatorial formula for macdonald polynomials 3 here and throughout. A combinatorial formula for macdonald polynomials request pdf. These formulas are generalizations of the formulas of haglundhaimanloehr for macdonald polynoimals of type gln. Read combinatorial formula for macdonald polynomials and generic macdonald polynomials, transformation groups on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. We describe two of them here, both of which ask to explain the symmetry between and exhibited by certain combinatorially defined polynomials in. In this interactive graph, you can see examples of polynomials with degree ranging from 1 to 8.
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. A new recursion for threecolumn combinatorial macdonald. At q t 0 the symmetric macdonald polynomials are the weyl characters or schur functions. Combinatorial formula for macdonald polynomials, bethe.
On combinatorial formulas for macdonald polynomials. The jack polynomial is a homogeneous, symmetric polynomial which generalizes the schur and zonal polynomials, and is in turn generalized by the heckmanopdam polynomials and macdonald polynomials. He later introduced a nonsymmetric generalization in 1995. Return the nonsymmetric macdonald polynomial in type a corresponding to a shape mu, with basement permuted according to pi. Haglundhaimanloehr hhl05b, hhl05a gave a combinatorial formula for the. A combinatorial formula for nonsymmetric macdonald 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. In this paper we use the combinatorics of alcove walks to give uniform combinatorial formulas for macdonald polynomials for all lie types. 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. If a is a polynomial or formal series, pka denotes the result of substituting ak for each. Publications and preprints sami assaf usc dana and david.
At q0 these formulas specialize to the formula of schwer for the macdonald spherical function in terms of positively folded. In order to master the techniques explained here it is vital that you undertake plenty of. 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. 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. These are the spherical functions for gk, where g is a padic. Polynomial functions mctypolynomial20091 many common functions are polynomial functions. The formula for macdonald polynomials which we give in theorem 3. If t1 the macdonald polynomials become the sums over w orbits, which are the monomial symmetric functions when the root system has type a.
A combinatorial model for the macdonald polynomials. Yip advances in mathematics 226 2011 309331 of intertwining operators i to the generator 1 of the polynomial representation of the daha see 6, prop. 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. The degree of a polynomial is the highest power of x that appears. In this thesis, we study the nonsymmetric macdonald polynomials e x. Expand function in legendre polynomials on the interval 1,1. Combinatorial formula for macdonald polynomials 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.