On D. E. Littlewood’s algebra of $S$-function
On D. E. Littlewood’s algebra of $S$-function
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关于 D. E. Littlewood 的 $S$ 函数代数
DOI：
10.1090/s0002-9939-1956-0078366-8
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发表时间：
1956
期刊：
影响因子：
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通讯作者：
E. M. Ibrahim
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文献类型：
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作者：
E. M. Ibrahim
因此[A(i]A=KxA，符号E2表示直和，At表示A的不变矩阵，对应于划分(X)=()2X2xi)。因此，Littlewood能够定义一个新的S函数乘法{X}X{(-ti=K，IV})，其中由0表示的运算称为S函数的乘法，而表达式{x}0{，}称为“X乘法u”。不变量理论的主要问题之一是求{i×J{，}其中(X)可以是(M)的一个划分，(A)n的一个划分。Littlewood是第一个用不同的方法成功地解决这个问题的人[7；8]。他最好的作品之一是“第三种方法”。他发现，如果{X}C){ni}=zv}，则Fgi，r{}-{}={n-1}*[Zegli，{x.}]其中Grat是由S函数的乘法通过{r)}{S}=gr{t{}t定义的。对于小度，如果Egir？(&)已知，则可以很容易地推断z{v}，这是{X}X0{n}所需的展开。随着度数的增加，某些选择就会出现，很难选择正确的{v i。然而，如果与其他一些方法结合使用，当出现困难时，这种方法可能仍然是有价值的，这些方法将表明正确的选择。后来对这个问题进行了几次攻击(Duncan[1；2]，Foulkes[3；4]，Murnaghan[9；10；11]，Newell[12]，Robinson[13；14]，Todd[16]，Thrall[15]，Zia-ud-Din[17])，在它们的一般性和所得结果适用于数值情况的程度上有很大的不同。在这篇文章中，使用了一个结合了
Thus [A ( I] A= KxA, the symbol E2 denoting direct sum and A t denoting the invariant matrix of A corresponding to the partition (X) = ()2X2 Xi) Hence Littlewood was able to define a new multiplication of Sfunctions {X} X {(-t I = K,, Iv} where the operation denoted by 0 is called the plethysm of S-function and the expression {x } 0 {, } is read "X plethys u." One of the main problems of the invariant theory is to evaluate {I X J {, } where (X) may be a partition of (m) and (A) a partition of n. Littlewood was first to tackle successfully this problem by applying different methods [7; 8]. One of his best is "the third method." He found that if {X}C){nI}=zv} then Fgi,r{ }-{ }= {n-1} * [ZEgli, {x. } ] where grat is defined from the multiplication of S-function by means of {r)} { s} =gr{t{}t. For small degrees if Egir?(&) is known then z { v } may be easily inferred and this is the required expansion of {X} X0 {n}. As the degrees get larger certain alternatives present themselves and it becomes difficult to select the correct {v I . However, the method may still be valuable if used in conjunction with some other method which would indicate the correct choice when difficulty arises. Later several attacks (Duncan [1; 2], Foulkes [3; 4], Murnaghan [9; 10; 11], Newell [12], Robinson [13; 14], Todd [16], Thrall [15], Zia-Ud-Din [17]) 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. In this paper the use of a general theorem taken in conjunction with