Counting characters of upper triangular groups
Counting characters of upper triangular groups
复制标题
计算上三角群的字符数
DOI：
10.1016/j.jalgebra.2007.01.027
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发表时间：
2007
影响因子：
0.9
通讯作者：
I. Isaacs
中科院分区：
文献类型：
--
作者：
I. Isaacs
在q阶有限域上，设Un（q）是对角元均为1的上三角n×n矩阵群.已知Un（q）的所有复不可约特征标的度数都是q的幂，且所有幂qe对0 ∈ e <$μ（n）都出现，其中上界μ（n）由公式μ（2 m）=m（m−1）和μ（2 m +1）=m2定义。证明了次数为qe的不可约特征标的个数是q中某个系数为整数的多项式（取决于n和e）。在本文中，我们为e是μ（n），μ（n）−1和1之一的情况构造了显式多项式。此外，还列出了一个多项式列表，这些多项式似乎给出了n 9的正确字符计数。也包括了一些有关模式群的结果。
Working over a finite field of order q, let Un(q) be the group of upper triangular n×n matrices with all diagonal entries equal to 1. It is known that all complex irreducible characters of Un(q) have degrees that are powers of q and that all powers qeoccur for 0⩽e⩽μ(n), where the upper bound μ(n) is defined by the formulas μ(2m)=m(m−1) and μ(2m+1)=m2. It has been conjectured that the number of irreducible characters having degree qeis some polynomial in q with integer coefficients (depending on n and e). In this paper, we construct explicit polynomials for the cases where e is one of μ(n), μ(n)−1 and 1. Also, a list of polynomials is presented that appear to give the correct character counts for n⩽9. Several related results concerning pattern groups are also included.