Modular Schur functions
Modular Schur functions
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模块化 Schur 函数
DOI：
10.1090/s0002-9947-1994-1273543-0
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发表时间：
1994
影响因子：
1.3
通讯作者：
G. Walker
中科院分区：
文献类型：
--
作者：
G. Walker
考虑了一类新的对称函数。这些函数类似于经典舒尔函数，但依赖于整数模p > 2，以及分拆x。在p为素数的情况下，这些函数中的某些函数在域K的自然特征p中是一般线性群GL（n， K）的不可约特征。这二重化了著名的由经典舒尔函数给出的詹姆斯的判据。在素数特征为p的代数闭域K上，一般线性群GL（n， K）的简单p模多项式特征的确定问题被认为是既重要又困难的问题。从某种意义上说，这是令人惊讶的，因为众所周知，如何显式地构造一个简单模块的完备集，或者作为Weyl模块V(X)的商，或者作为它们的对偶的子模块，即相应的Schur模块H°(X)。（最近的论述见[14，§3]）困难在于从这些一般结构中提取字符的公式，甚至计算模块的尺寸。对于对称群的简单模表示，也出现了同样的情况；事实上，g.d.詹姆斯已经证明，这两个问题实际上是等价的。然而，这个问题的一些特殊情况是可以处理的。权重X最高的Weyl和Schur模块的（正式）特征是Schur函数Sx，对于特征为0的域K，这些是简单的字符。James和Murphy[6,9]给出了对于所有足够大的n， Sx是GL（n， K）的一个简单模特征的充分必要条件（最先由R. W. Carter推测）。本文的目的是证明，对于奇素数p，存在第二组简单模字符的自然一般公式，也由满足Carter准则的分区索引，并确定这些简单字符的最高权重。主要结果是定理4.4。结果对于p = 2也是有效的，但是在这种情况下没有给出新的信息。这些简单的字符属于对称函数族，我们称之为模舒尔函数。正如在[4，§3]中所解释的，我们考虑对称函数a的环的自同态co'，它是在生成子hj上定义的，它是d次的完全对称函数，在变量的p次幂处截断。这里p可以是任意整数>2，不一定是素数。根据定义，图像co'(f) = f（/）是“模”对称的。1991数学学科分类。初级05E05, 20C20；二级05E10, 20C30。©1994美国数学学会0002-9947/94 $1.00+ $。许可证或版权限制可能适用于再分发；参见http://www.ams.org/journal-terms-of-use
A new family of symmetric functions is considered. These functions are analogous to the classical Schur functions, but depend on an integer modulus p > 2 , as well as on a partition X. In the case where p is prime, certain of these functions are shown to be irreducible characters of the general linear group GL(n , K) in the natural characteristic p of the field K . This dualises a wellknown criterion of G. D. James for such characters to be given by classical Schur functions. The problem of determining the simple p-modular polynomial characters of the general linear group GL(n, K) over an algebraically closed field K of prime characteristic p is (justly) regarded as both important and very difficult. In one sense this is surprising, because it is well known how to construct a complete set of simple modules explicitly, either as quotients of Weyl modules V(X), or as submodules of their duals, the corresponding Schur modules H°(X). (A recent treatment is given in [14, §3].) The difficulty lies in extracting formulae for the characters, or even in calculating the dimensions of the modules, from these general constructions. The same situation arises with regard to the simple modular representations of the symmetric group; and indeed, G. D. James has shown that the two problems are, in effect, equivalent. Some special cases of this problem are, however, tractable. The (formal) character of the Weyl and Schur modules with highest weight X is the Schur function Sx , and for a field K of characteristic 0 these are the simple characters. James and Murphy [6, 9] have given a necessary and sufficient condition (first conjectured by R. W. Carter) for Sx to be a simple modular character of GL(n, K) for all sufficiently large n . The purpose of this article is to show that, for odd primes p , there is a natural general formula for a second family of simple modular characters, also indexed by partitions satisfying the Carter criterion, and to determine the highest weight of these simple characters. The main result is Theorem 4.4. The results are valid also for p = 2, but give no new information in this case. These simple characters belong to a family of symmetric functions which we call modular Schur functions. As explained in [4, §3], we consider an endomorphism co' of the ring of symmetric functions A. This is defined on the generator hj , the complete symmetric function of degree d, by truncation at the pth powers of the variables. Here p can be any integer > 2, not necessarily prime. The image co'(f) = f of / is, by definition, the 'modular' symmetric Received by the editors November 21, 1993. 1991 Mathematics Subject Classification. Primary 05E05, 20C20; Secondary 05E10, 20C30. © 1994 American Mathematical Society 0002-9947/94 $1.00+ $.25 per page 569 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use