Representations of the symmetric groups over the field of order 2
Representations of the symmetric groups over the field of order 2
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2 阶域上对称群的表示
DOI：
10.1016/0021-8693(76)90220-9
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发表时间：
1976
影响因子：
0.9
通讯作者：
G. James
中科院分区：
文献类型：
--
作者：
G. James
罗宾逊在其著作[3]中给出了一种方法，确定了奇素数p的对称群的所有p-模不可约特征标，但他对素数2的结果是不正确的。在[4]中，他解释了为什么他的方法是错误的，并给出了S的修正的2-模分解矩阵，直到n= 8和S的分解矩阵，他认为这是正确的。他的方法非常费力，实际上不可能把结果推得更远。在本文中，这个问题是从一个完全不同的观点。对于所有对称群，找到了分解矩阵中对应于[n-m，m]和[n-m-1，m，I]型特征标的部分.虽然当这里使用的方法扩展到其他类型的字符时，复杂性会成倍增加，但毫无疑问，它们可以用来确定任何特定对称群的所有2-模表示。在我们考虑的情况下，从一个普通特征标x出现的2-模不可约特征标被发现是一个模的模特征标，其2-模特征标是X，模一个自然双线性型的核。这两种类型所考虑的另一个共同特征是所涉及的分解矩阵的部分非常相似。利用[1]中的模不可约性知识和方法，可以从本文给出的结果中找到更多的模特征标。这在最后一节中说明，其中检查了罗宾逊关于Ss和S1的结果，并找到了S1的所有模不可约数。
In his book [3], Robinson gave a method for determining all thep-modular irreducible characters of the symmetric groups for odd primes p. However, his results for the prime 2 were incorrect. In [4], he gave an explanation of why his methods were wrong and gave amended 2-modular decomposition matrices for S, up to n= 8 and a decomposition matrix for S’s which he believed to be correct. His method is so laborious that it is, in practice, impossible to push the results much further. In this paper, the problem is approached from an entirely different point of view. The parts of the decomposition matrices corresponding to characters of types [n-m, m] and [n-m-1, m, I] are found for all symmetric groups. Although the complications will multiply when the methods used here are extended to characters of other types, undoubtedly they can be used to determine all the 2-modular representations of any particular symmetric group. In the cases we consider, the 2-modular irreducible character turning up from an ordinary character x is found to be the modular character of a module, whose 2-modular character is X, modulo the kernel of a natural bilinear form. Another feature common to both types considered is that the parts of the decomposition matrices involved are very similar. More modular characters of S, can be found from the results given here, by using the knowledge of modular irreducibles of S,-, and the methods used in [l]. This is illustrated in the final section, where Robinson’s results for Ss and S, are checked and all the modular irreducibles of S,, are found.