The Combinatorics of Macdonald Polynomials

PI计划从事三个相关项目，涉及麦克唐纳多项式的组合学和对角谐波空间。 第一个是建立在最近的联合工作之间的Haiman，Loehr和PI的组合公式的A型非对称Macdonald多项式。在A型设置这个公式的存在表明，类似的公式存在于其他根系;发现这样的公式将打开主题显着，因为很少有明确的身份已经发现任何类型的A型以外的情况。 第二个项目试图建立在（A型）对称Macdonald多项式的组合公式上，该公式由PI根据经验发现，并在随后与Haiman和Loehr的联合工作中得到证明。 在这方面的主要开放问题是找到一个组合公式的系数在舒尔展开;作为一个步骤，在这个方向上的PI描述了一个新的简化公式的特殊情况下增广hook shapes.The第三个项目涉及试图证明一个简化公式的单项式扩展的字符的空间的对角谐波，由于海曼，Loehr，Remmel，Ulyanov和PI。 这一猜想使PI得出了上述Macdonald多项式的经验公式，它对计数组合学和表示理论有许多影响。正交多项式是满足正交条件的多项式族，通常意味着族中任何两个不同元素的积分，对某个给定的权重函数，为零。正交多项式的研究已有几个世纪的历史，它们在数学和科学中有着相当多的应用。对称函数是多元的多项式，它在变量的任何排列下都是不变的。 在1988年Macdonald提出了一个新的正交多项式族，它依赖于一组变量X和两个额外的参数q，t。 它们是变量X中的对称函数，并且包含作为特殊情况的大多数有用的对称函数。它们立即被认为对数学的几个领域很重要，包括组合学和特殊函数。在2000年，海曼证明了它们在被称为代数几何的数学的高级、抽象和理论分支方面有复杂的解释。然而，麦克唐纳的构造是间接的，最近PI发现了Macdonald多项式的一个简单的组合公式，并由Haiman和Loehr证明。 PI正试图应用这一结果来回答围绕它们的一些开放问题。Macdonald和Cherednik现在有一个更一般的结构，它包含了几乎所有以前研究的对称函数和正交多项式作为特殊情况。PI与Haiman合作，正试图将简单的组合公式扩展到这个更一般的结构。

The PI plans to work on three related projects involving the combinatoricsof Macdonald polynomials and the space of diagonal harmonics. The firstis an attemptto build on recent joint work between Haiman, Loehr and the PI which gives acombinatorial formula for type A nonsymmetric Macdonald polynomials.The existence of this formula in the type A setting suggests that similarformulas exist for other root systems; the discovery of such formulaswould open up the subject significantly, as very few explicit identities ofany kind have been discovered beyond the type A case. The second projectattempts to build onthe combinatorial formula for (type A) symmetric Macdonald polynomials,which was discovered empirically by the PI and proved in subsequent jointworkwith Haiman and Loehr. The main open problem in this regard is to finda combinatorial formula for the coefficients in theSchur expansion; as a step in this directionthe PI describes a new conjectured formula for the special case ofaugmented hook shapes.The third project involves trying to prove a conjecturedformula for the monomial expansion of thecharacter of the space of diagonal harmonics, due to Haiman, Loehr, Remmel,Ulyanov and the PI. This conjecture,which led the PI to the empirical formula for symmetricMacdonald polynomials mentioned above,has a number of implications to enumerative combinatorics andrepresentation theory.Orthogonal polynomials are families of polynomials which satisfy anorthogonality condition, typically meaning that the integral of anytwo different elements of the family,against some given weight function, is zero.The study of orthogonal polynomials is several centuries old,and they have quite a number of applications throughout mathematics andscience.Symmetric functions are polynomials in several variables which areinvariant under any permutation of the variables. They have a large numberof applications to several branches of mathematics such as representationtheory andthe roots of polynomial equations.In 1988 Macdonald introduced a new family of orthogonal polynomials,which depend on a set of variables X and two extra parameters q,t. They aresymmetric functions in the variables X, andcontain most useful symmetric functions as special cases.They were immediately recognized as being important to several area ofmathematicsincluding combinatorics and special functions.In 2000 Haiman proved they have a complicated interpretation in terms ofan advanced,abstract and theoretical branch ofmathematics known as algebraic geometry.Macdonald's construction is indirect however, and neither it norHaiman's interpretationgive a way of easily describing these polynomials.Recently the PI discovered, and with Haiman and Loehr proved, a simplecombinatorial formula for Macdonald's polynomials. The PI is trying toapply this result to answer some of the open questions surrounding them.Macdonald and Cherednik now have a much more general construction whichcontains almost allpreviously studied symmetric functions and orthogonal polynomials asspecial cases.The PI, in collaboration with Haiman, is trying to extend thesimple combinatorial formula to this more general construction.