Basic Representations ofA(1)n−1andA(2)2nand the Combinatorics of Partitions☆
Basic Representations ofA(1)n−1andA(2)2nand the Combinatorics of Partitions☆
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A(1)n−1 和 A(2)2n 的基本表示以及划分的组合☆
DOI：
10.1006/aima.1998.1781
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发表时间：
1999
影响因子：
1.7
通讯作者：
S. Leidwanger
中科院分区：
文献类型：
--
作者：
S. Leidwanger
在划分的组合学和仿射李代数的表示理论之间有一个很好的联系(例如见[7，20，14]，以及[13]中的一些参考文献)。特别地，已经证明了A、D、E型仿射李代数的一级表示的弦函数有一个关于经典配分函数的简单表达式(见[13]，第二章)。12、13)。类似地，这些表示的特征的主要专门化是满足一些简单约束的分割数的生成序列(见[13]，第14.4节。4)。其原因可以在Frenkel Kac[6]在齐次分次情形下和Kac Kazhdan Lepowsky Wilson[16]在主分次情形下通过作用于多项式环上的微分算子来显式实现水平L表示中找到。在40年代的S中，中山提出了关于群Sm的p-块的一个著名猜想，它的证明最早是由Brauer和Robinson给出的。然后，Robinson发展了对称群的模表示理论。他引入了划分的星图(也称为p-商)，并证明了划分完全由它的p-核和p-商决定。这使他能够计算属于给定块的普通不可约字符和模不可约字符的数量[31]。Robinson还定义了一些精化的归纳和限制算子(称为r-限制和r-归纳，其中r=0，…，p&1)，并用它们计算对称群的分解矩阵。
There is a well-established connection between the combinatorics of partitions and the representation theory of affine Lie algebras (see for example [7, 20, 14], and a number of references in [13]). In particular, it has been proved that the string functions of the level 1 representations of the affine Lie algebras of type A, D, E have a simple expression in terms of the classical partition function (see [13], Chap. 12, 13). Similarly, the principal specialization of the character of these representations is the generating series of the number of partitions satisfying some simple constraints (see [13], Sect. 14.4. 4). The reason for that may be found in the explicit realizations of the level l representations by means of differential operators acting on polynomial rings, given by Frenkel Kac [6] in the homogeneous grading case and by Kac Kazhdan Lepowsky Wilson [16] in the principal grading case.Since the works of Frobenius and Young, the combinatorics of partitions is also well known to be related to the representation theory of the symmetric groups. In the 40's, Nakayama formulated a famous conjecture concerning the p-blocks of the groups Sm whose proof was first given by Brauer and Robinson. The modular representation theory of the symmetric groups was then developed by Robinson. He introduced the star diagram of a partition (also called the p-quotient) and showed that a partition is completely determined by its p-core and p-quotient. This allowed him to compute the number of ordinary and modular irreducible characters pertaining to a given block [31]. Robinson also defined some refined induction and restriction operators (called r-restriction and r-induction where r= 0,..., p&1) and used them to calculate decomposition matrices of symmetric groups.