Combinatorial Representation Theory which Center of Schur Functions

这是为了理解舒尔函数和舒尔Q函数(舒尔函数的射影模拟)在表示论中的作用。更准确地说，我们证明了下面的定理。与矩形Young图相关的Schur函数是D^{(2)}_2型仿射李代数的基本表示的权重向量，也是非线性薛定谔族的齐次tau函数。证明这一点的关键思想是用费米子写下表示空间和算符，并通过玻色子-费米子对应导出多项式。我们已经成功地证明了A型仿射李代数的类似现象。2004年，我们考虑了以下问题。阐明了用舒尔Q函数展开时，2个约化舒尔函数中系数的性质。通过一些小秩次情况下的实验计算，我确信这些系数无论是从表示理论还是从组合的角度来看都是非常有意义的。最后，我们意识到这些系数只不过是所谓的斯坦布里奇数。因此，我们可以将这些数字与仿射李代数的表示理论联系起来。仔细查看这些数字的表，我们发现了对称群的Cartan矩阵的初等因子的一个简单公式。10多年前，奥尔森在哥本哈根给出了一个公式，它是用母函数表示的，而且相当复杂。我们的版本更直接、更具组合性。

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.