Plethysm of S-Functions
Plethysm of S-Functions
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S 函数的丰富性
DOI：
10.1098/rsta.1954.0008
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发表时间：
1954
期刊：
影响因子：
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通讯作者：
H. O. Foulkes
中科院分区：
文献类型：
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作者：
H. O. Foulkes
将不变矩阵的不变矩阵表示为不可约不变矩阵的直和的问题是将两个S-函数的体积m {d} O {u}表示为S-函数的和的问题。几个攻击（Littlewood 1944，1951; Murnaghan 1951 a，B，c;罗宾逊1949，1950; Thrall 1942;托德1949）已经对这个问题，变化很大，在其普遍性和程度上所获得的结果已适用于数值情况。随着{d}和{/U>}的权重的增加，越来越明显的是，需要开发一种在体积描记中找到任何给定S函数的系数的技术，而不是一种确定完全展开的方法，这种方法不可避免地非常费力。此外，需要考虑任何给定体积的结构，以查看这些项是否可以以任何系统的方式被分组为集合，或者所产生的系数之间是否存在任何关系。这类情况已知是{m}O Sr和{lm}® Sr的情况（Foulkes 1951），其中后一个结果中出现的具有k个部分的每个S-函数可以由m的一个划分以及k个非负整数的集合来表征。
The problem of expressing an invariant matrix of an invariant matrix as a direct sum of irreducible invariant matrices is that of expressing the plethysm {d} O {u} of two S-functions as a sum of S-functions. Several attacks (Littlewood 1944, 1951; Murnaghan 1951 a, b, c; Robinson 1949, 1950; Thrall 1942; Todd 1949) have been made on this problem, varying considerably in their generality and the degree to which the results obtained have been applicable to numerical cases. As the weights of {d} and {/U>} increase it becomes more and more apparent that there is need to develop a technique of finding the coefficient of any given S-function in a plethysm, rather than a method, inevitably very laborious, of determining the full expansion. Furthermore, there is need to consider the structure of any given plethysm to see whether the terms can be grouped into sets in any systematic way, or whether any relations exist between the coefficients that arise. Something of this sort is known to be the case for {m}O Sr and {lm}® Sr (Foulkes 1951), in which every S-function with k parts appearing in the latter result can be characterized by a partition of m together with a set of k non-negative integers.