At q0 these formulas specialize to the formula of schwer for the macdonald spherical function in terms of positively folded. Combinatorial definition of halllittlewood polynomials. A combinatorial model for the macdonald polynomials. Publications and preprints sami assaf usc dana and david. In our setting, the charge, an intricate combinatorial statistic appearing. On combinatorial formulas for macdonald polynomials. Note that this is a different reading order than the usual french or japanese ones. The form of these generic polynomials is that of a bethe eigenfunction and they imitate, on a more elementary level, the r. He later introduced a nonsymmetric generalization in 1995. Combinatorics of macdonald polynomials and cyclic sieving. We describe two of them here, both of which ask to explain the symmetry between and exhibited by certain combinatorially defined polynomials in.
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 main application of schur polynomials in representation theory of the. 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. In this thesis, we study the nonsymmetric macdonald polynomials e x. We further introduce a general family of polynomials f t z. 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. 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. Citeseerx document details isaac councill, lee giles, pradeep teregowda. At q t 0 the symmetric macdonald polynomials are the weyl characters or schur functions. 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 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. Jack symmetric functions and some combinatorial properties of young symmetrizers. A combinatorial model for the macdonald polynomials pnas.
A combinatorial formula for macdonald polynomials arxiv. Since the former is given by halllittlewood polynomials, we arrive at a combinatorial expression of halllittlewood polynomials as above. 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. Cauchy in, where he defined the schur functions as a ratio of alternants, and proving that these are symmetric. In mathematics, the jack function is a generalization of the jack polynomial, introduced by henry jack. A combinatorial formula for macdonald polynomials sciencedirect.
Jacobi gave the definition of schur polynomials via cauchys bialternant formula and proved the jacobitrudi identity. Polynomial functions and equations what is a polynomial. Combinatorial formula for macdonald polynomials and. Of recent note in the theory of macdonald polynomials has been the. The jack polynomials are a family of symmetric functions which extends the schur polynomials. Polynomial functions mctypolynomial20091 many common functions are polynomial functions.
These developments led in turn to some surprising discoveries in the combinatorics of macdonald polynomials, which are described in appendix a. They are indexed by integer partitions and constitute a basis for the space of symmetric functions. A combinatorial formula for macdonald polynomials request pdf. Haglundhaimanloehr hhl05b, hhl05a gave a combinatorial formula for the. 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 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. The partitions of a given n are partially ordered by 3. He also posed the problem of finding a combinatorial rule to describe these polynomials.
A new recursion for threecolumn combinatorial macdonald polynomials elizabeth niese dept. It is also possible to get a formula as sum over ssyts and the cocharge statistic. Combinatorial formula for macdonald polynomials and generic. We prove a combinatorial formula for the macdonald polynomial. We give a combinatorial proof of this conjecture, which establishes the existence and integrality of h. Before we look at the formal definition of a polynomial, lets have a look at some graphical examples.
In this interactive graph, you can see examples of polynomials with degree ranging from 1 to 8. 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. A combinatorial formula for nonsymmetric macdonald. Two q,tsymmetry problems in symmetric function theory. In this paper we use the combinatorics of alcove walks to give uniform combinatorial formulas for macdonald polynomials for all lie types. A combinatorial expression of halllittlewood polynomials. Haglund recently proposed a combinatorial interpretation of the modified macdonald polynomials h. Sahi and give two explicit formulas for these polynomials.
If a is a polynomial or formal series, pka denotes the result of substituting ak for each. Polynomial functions mcty polynomial 20091 many common functions are polynomial functions. The degree of a polynomial is the highest power of x that appears. Combinatorial formulas for macdonald and halllittlewood. 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. Binomial formula for macdonald polynomials and its applications andrei okounkov abstract. A combinatorial formula for macdonald polynomials 3 here and throughout. 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. 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. Just because this is a statement about the polynomials themselves rather than symmetric functions. Note that if both \q\ and \t\ are specified, then they must have the same parent.
From their definition, all one can infer is that the k. This implies the schur positivity of the kribbon tableaux generating functions introduced by lascoux, leclerc and thibon. 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 recursion and a combinatorial formula for jack polynomials. Nov 16, 2004 he also posed the problem of finding a combinatorial rule to describe these polynomials. These formulas are generalizations of the formulas of haglundhaimanloehr for macdonald polynoimals of type gln. 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. These are the spherical functions for gk, where g is a padic. In this paper we use the combinatorics of alcove walks to give a uniform combinatorial formula for macdonald polynomials for all lie types. On a positivity property of halllittlewood polynomials.
These formulas are sums over llings of tableaulike diagrams. This book explains some recent applications of the theory of polynomials and algebraic geometry to combinatorics and other areas of mathematics. In this case our formula for the macdonald polynomial specializes to the formula for the weyl character in terms of the. These formulas resemble the formulas of haglund, haiman and loehr for macdonald polynomials of type gl n. Publications and preprints sami assaf usc dana and. Ram and yip gave a formula for the macdonald polynomials of arbitrary type in terms of socalled alcove walks. Our main tool is a qintegral representation for ordinary macdonald polynomial. In this unit we describe polynomial functions and look at some of their properties.
A combinatorial formula for macdonald polynomials citeseerx. When two polynomials are divided it is called a rational expression. In other words, we have been calculating with various polynomials all along. In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature. Pdf a combinatorial formula for macdonald polynomials. Combinatorial definition of halllittlewood polynomials sum. We generalize the binomial formula for jack polynomials proved in oo2 and consider some applications. A combinatorial formula for nonsymmetric macdonald polynomials. Macdonald originally associated his polynomials with weights. At q 0 the symmetric macdonald polynomials are the halllittlewood polynomials or the macdonald spherical functions. Corollaries to our main theorem include the expansion of. 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. Combinatorial formula for macdonald polynomials, bethe ansatz. Combinatorial formula for macdonald polynomials, bethe.
The formula for macdonald polynomials which we give in theorem 3. 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. In our setting, the charge, an intricate combinatorial statistic appearing in the lascouxsch. 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. The formula generalizes our previous combinatorial interpretation. There has been a lot of work devoted to understanding macdonald polynomials from a combinatorial point of view. A new recursion for threecolumn combinatorial macdonald. In order to master the techniques explained here it is vital that you undertake plenty of. Return the nonsymmetric macdonald polynomial in type a corresponding to a shape mu, with basement permuted according to pi. Relationship between legendre polynomials and legendre functions of the second kind. A polynomial is a mathematical expression constructed with constants and variables using the four operations.