On mixed products of complex characters of the double covers of the symmetric groups
On mixed products of complex characters of the double covers of the symmetric groups
复制标题
对称群双覆盖复性混合积的研究
DOI：
--
复制
发表时间：
2001
期刊：
影响因子：
--
通讯作者：
C. Bessenrodt
中科院分区：
文献类型：
--
作者：
C. Bessenrodt
对称群S_n的复特征标的Kronecker积在许多文献中都有研究。关于特殊乘积和特殊成分系数的信息已经获得，但目前还没有有效的组合算法来计算这些乘积。在文献[1]中，给出了交错群An的特征标具有较少齐次成分的乘积和特征标的齐次积的分类。特别地，对于Sn，不存在非平凡的齐次Kronecker积，但对于An，当n是平方数时，存在这样的积(这些积甚至是不可约的)。对于对称群的双重覆盖S̃n，关于特征标乘积的信息更加稀疏。最近，在[2]中，关于S̃n的自旋特征积得到了一些结果，从而导致了对均匀自旋产物的分类。这里，对所有的三角数n都有齐次积，但只有当n=6时才有非平凡的不可约积。本文考虑双重覆盖S̃n的复特征标的混合积，即S̃n的一个非忠实特征标(对应于一个n的特征标)与一个自旋特征标的乘积。在这种情况下，有一些有趣的齐次或几乎齐次的混合乘积；John Stembridge用来处理对称函数的特殊枫包SF和QF极大地帮助了这种混合乘积的找到。两个齐次族分别。描述了几乎齐次的乘积；一个用于任意复合数，另一个用于三角数。然后对不可约混合积进行分类；它们出现在满足同余条件的偶数和三角数中。
Kronecker products of complex characters of the symmetric group Sn have been studied in many papers. Information on special products and on the coefficients of special constituents have been obtained but there is no efficient combinatorial algorithm in sight for computing these products. In [1], products of Sn-characters with few homogeneous components and homogeneous products of characters of the alternating group An have been classified. In particular, there are no non-trivial homogeneous Kronecker products for Sn, but there are such products for An, when n is a square number (these are even irreducible). For the double covers S̃n of the symmetric groups, information about products of characters is even more sparse. Recently, in [2] some results have been obtained on products of spin characters of S̃n which led to a classification of homogeneous spin products. Here, homogeneous products do occur for all triangular numbers n, but non-trivial irreducible products occur only for n = 6. In this article, we consider mixed products of complex characters for the double covers S̃n, i.e., products of a non-faithful character of S̃n (corresponding to a character of Sn) with a spin character. In this situation, there are some interesting homogeneous or almost homogeneous mixed products; finding such mixed products was greatly helped by the special maple packages SF and QF for dealing with symmetric functions by John Stembridge. Two families of homogeneous resp. almost homogeneous products are described; one for any composite number and the other one for triangular numbers. The irreducible mixed products are then classified; they occur for even numbers and triangular numbers satisfying a congruence condition.