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Combinatorial Representation Theory which Center of Schur Functions

以Schur函数为中心的组合表示论

基本信息

批准号：
15540030
负责人：
YAMADA Hirofumi
金额：
$ 1.86万
依托单位：
OKAYAMA UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2004
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540030/
关键词：
Schur functions Symmetric groups affine Lie algebras ヒェルルヒー

项目摘要

This is an effort for understanding the role of Schur functions and Schur's Q functions, the projective analogue of Schur functions, in representation theory. To be more precise, we proved the following theorem. Schur functions associated with the rectangular Young diagrams occur as weight vectors of the basic representation of the affine Lie algebra of type D^{(2)}_2. And also, they turn out to be the homogeneous tau functions of the nonlinear Schroedinger hierarchy. The key idea for proving the above is to write down the representation spaces and operators in terms of fermions, and derive polynomials via the boson-fermion correspondence. We have succeeded in verifying the similar phenomena for the case of the affine Lie algebra of type A^{(2)}_2. In 2004 we considered the following problem. Clarify the nature of the coefficients in the 2 reduced Schur functions when expanded in terms of Schur's Q functions. Through some experimental computations in small rank cases, I had been convinced that these coefficients are of great interests, both from representation theoretical and combinatorial points of view. Finally we realized that these coefficients are nothing but the so-called Stembridge numbers. As a result we could relate these numbers with the representation theory of affine Lie algebras. Looking carefully at the table of these numbers, we found a simple formula for the elementary divisors of the Cartan matrices of the symmetric groups. More than 10 years ago, Olsson in Copenhagen gave a formula for those, which is expressed in terms of a generating function and is rather complicated. Our version is more direct and combinatorial.

这是一个努力理解的作用舒尔职能和舒尔的Q职能，投影模拟舒尔职能，在表示论。为了更准确地说明，我们证明了下面的定理。与矩形Young图相关的Schur函数作为D^{（2）}_2型仿射李代数的基本表示的权向量出现。同时，它们也是非线性薛定谔族的齐次τ函数。证明上述问题的关键思想是用费米子来表示表示空间和算符，并通过玻色子-费米子对应导出多项式。我们成功地证明了A^{（2）}_2型仿射李代数的类似现象。2004年，我们考虑了以下问题。阐明了在2个约化Schur函数中的系数在Schur的Q函数方面展开时的性质。通过在小秩情况下的一些实验计算，我已经确信，这些系数是极大的兴趣，无论是从表示理论和组合的观点。最后，我们意识到这些系数只不过是所谓的Stembridge数。因此，我们可以将这些数与仿射李代数的表示理论联系起来。仔细观察这些数字的表，我们发现了对称群的Cartan矩阵的基本因子的简单公式。10多年前，Olsson在哥本哈根给出了一个公式，这个公式是用母函数表示的，相当复杂。我们的版本更直接和组合。