Decomposition Numbers for Symmetric Groups and Composition Factors of Weyl Modules
Decomposition Numbers for Symmetric Groups and Composition Factors of Weyl Modules
复制标题
对称群的分解数和 Weyl 模的组成因子
DOI：
--
复制
发表时间：
1996
期刊：
影响因子：
--
通讯作者：
K. Erdmann
中科院分区：
文献类型：
--
作者：
K. Erdmann
假设S是对称群的程度r和K是一个代数r闭领域的主要特征p.一个主要问题的表示理论的S在K是理解分解r号码，即多重的组成因素的Specht模块。- 是的有一个类似的问题，多项式表示GL K，n即理解的组成因素的Weyl模块。这似乎是一个难题，而A型的Lusztig猜想给出了答案。Ja发现对称群S的分解数与Weyl r的复合重数相同。GL K的模，如果r s n，反之，大多数组合n ≠ n。由S的分解数可以求出齐次为n的GL K的Weyl模的因子.我们将在这里证明，事实上所有的合成重Weyl模等于分解数的对称群，如果一个允许r变化。事实上，有精确的公式与组成有关。多重性见1.4。这些都是已知的几年;他们证明了使用倾斜模和理论的准遗传代数wx。参见R; D; E。在这里，我们观察到，一个特殊的情况可以用来获得在2.4中给出的结果。
Suppose S is the symmetric group of degree r and K is an algebraically r closed field of prime characteristic p. A major problem for the representation theory of S over K is that of understanding the decomposition r numbers, that is the multiplicities of the composition factors of the Specht modules. Ž . There is an analogue problem for polynomial representations of GL K , n namely that of understanding composition factors of Weyl modules. This appears to be a hard problem, and an answer to this is suggested by the Lusztig conjecture, for type A. w x It was discovered by Ja that the decomposition numbers for the symmetric group S are the same as composition multiplicities for Weyl r Ž . modules of GL K if r s n and that, conversely, most composition n Ž . factors of the Weyl modules of GL K of homogeneous degree n can be n calculated from decomposition numbers of S . n We shall prove here that in fact all composition multiplicities of Weyl modules are equal to decomposition numbers of symmetric groups if one allows r to vary. There are in fact precise formulae relating the composiŽ . tion multiplicities see 1.4 . These are known for a few years; they were proved using tilting modules and the theory of quasi-hereditary algebras Ž w x. see R; D; E . Here we observe that a particular case can be used to obtain the result which is given in 2.4.