A Supercharacter Table Decomposition via Power-Sum Symmetric Functions
A Supercharacter Table Decomposition via Power-Sum Symmetric Functions
复制标题
基于幂和对称函数的超级字符表分解
DOI：
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发表时间：
2011
影响因子：
0.8
通讯作者：
N. Thiem
中科院分区：
文献类型：
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作者：
N. Bergeron;N. Thiem
我们给出了n个群的超特征标表的AB-因子分解 在$FF_q$上的n$幂单上三角矩阵，其中$A$是元素在$中的下三角矩阵。Z[q]$和$B$是幂单上三角矩阵，元素在$中Z[q^{-1}]$。为此，我们引入了一个新的幂和基的对称函数在非交换变量的Hopf代数的q变形。利用超特征基、q幂和基与超类基之间的过渡矩阵，得到了分解式。这类似于对称群S_n的特征标表由Schur函数、单项式和幂和之间的转移矩阵所给出的分解。
我们推导出一些组合的结果与此分解。特别地，我们计算超字符表的行列式。
We give an $AB$-factorization of the supercharacter table of the group of $n imes n$ unipotent upper triangular matrices over $FF_q$, where $A$ is a lower-triangular matrix with entries in $Z[q]$ and $B$ is a unipotent upper-triangular matrix with entries in $Z[q^{-1}]$. To this end we introduce a $q$ deformation of a new power-sum basis of the Hopf algebra of symmetric functions in noncommutative variables. The factorization is obtain from the transition matrices between the supercharacter basis, the $q$-power-sum basis and the superclass basis. This is similar to the decomposition of the character table of the symmetric group $S_n$ given by the transition matrices between Schur functions, monomials and power-sums.
We deduce some combinatorial results associated to this decomposition. In particular we compute the determinant of the supercharacter table.