The Combinatorics of Macdonald Polynomials

国际和平研究所计划开展三个相关项目，涉及麦克唐纳多项式的组合和对角调和空间。首先试图建立在Haiman，Loehr和PI最近的联合工作的基础上，给出了A型非对称Macdonald多项式的一个组合公式。该公式在A型环境中的存在表明，对于其他的根系统也存在类似的公式；这类公式的发现将极大地开拓这一主题，因为除了A型情形之外，几乎没有发现任何类型的显式恒等式。第二个项目试图建立(A型)对称麦克唐纳多项式的组合公式，该公式是由PI经验性地发现的，并在随后与Haiman和Loehr的合作中得到证明。这方面的主要问题是找出舒尔展开式中系数的组合公式；作为这一方向的一个步骤，PI描述了关于增广钩形状的特殊情况的新的猜想公式。第三个项目涉及试图证明由于Haiman，Loehr，Remmel，Ulyanov和PI而导致的对角调和空间的特征的单项展开式的猜想公式。这一猜想将PI引向上述对称麦克唐纳多项式的经验公式，对计数组合学和表示理论有许多启示。正交多项式是满足正交性条件的多项式族，通常意味着该族中任何两个不同元素相对于某个给定权函数的积分为零。对正交多项式的研究已有数百年的历史，它们在整个数学和科学中有相当多的应用。对称函数是多个变量的多项式，在变量的任何排列下都是不变的。它们在数学的几个分支中有大量的应用，如表示理论和多项式方程的根。1988年，麦克唐纳引入了一族新的正交多项式，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，并且包含最有用的对称函数作为特例。它们立即被认为对包括组合学和特殊函数在内的几个数学领域是重要的。2000年，海曼证明了它们有一个被称为代数几何的高级、抽象和理论分支的复杂解释。然而，麦克唐纳的构造是间接的，最近PI发现并证明了麦克唐纳多项式的一个简单的组合公式。PI正试图应用这一结果来回答围绕他们的一些开放问题。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过的对称函数和正交多项式的特例。PI与Haiman合作，试图将简单的组合公式扩展到这种更一般的结构。

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.