Combinatorial aspects of representations of groups and algebras

我主要研究了Schur的Q-函数与仿射李代数之间的关系。首先，我发现q-函数，表示为幂和对称函数的多项式，构成了在多项式环上实现的某些仿射李代数的基本表示的权基。Q-函数通过严格的划分被参数化。利用杨图的一些组合方法，确定了给定Q-函数的权重。将这一过程应用于最简单的仿射李代数$A^{(1)}_1，找出由Schur函数和Q-函数满足的恒等式。起初，这个恒等式似乎很有趣：然而，利用对称群的自旋表示的分解矩阵证明了这一点。由于这一事实，我转向对分解矩阵本身的研究。作为第一个结果，我证明了当特征等于2时，自旋表示的分解矩阵的行列式等于2的幂。我研究的另一个特点是所谓的复反射群G(r，p，n)的“高光谱多项式”。群G(r，p，n)作用于n个变量的多项式环。“余不变环”是群上不变量生成的理想的商。已知G(r，p，n)在这个余不变环上的作用与正则表示同构。高阶Speht多项式自然地表现为每个不可约分量的基向量。

I focused on a relationship of Schur's Q-functions and affine Lie algebras. First I found that the Q-functions, expressed as polynomials of power sum symmetric functions, form a weight basis for the basic representation of certain affine Lie algebras, realized on a polynomial ring. Q-functions are parametrized by the strict partitions. Using some combinatorics of Young diagrams, I determined the weight of the given Q-function. This procedure was applied to the simplest affine lie algebra $A^{(1)}_1$ to find an identity satisfied by Schur functions and Q-functions indexed by some specific partitions. At first this identity seemed funny : However this was proved to be true by making use of decomposition matrices of the spin representations of the symmetric group. By virtue of this fact, I turned to a study of the decomposition matrices themselves. As a first result I proved that the determinant of the decomposition matrix of the spin representations is equal to a power of two when the characteristic equals two.Another feature of my research is the so called "higher Specht polynomials" for the complex reflection group G(r, p, n). The group G(r, p, n) acts on the polynomial ring of n variables. The "coinvariant ring" is the quotient by the ideal which is generated by invariants over the group. It is known that the action of G(r, p, n) on this coinvariant ring is isomorphic to the regular representation. The higher Specht polynomials appear naturally as basis vectors of each irreducible component.