论文标题
计数上部unitriangular群体中小度的字符
Counting characters of small degree in upper unitriangular groups
论文作者
论文摘要
令$ u_n $表示固定有限字段$ \ mathbb {f} $ of订单$ q $的上限$ n \ times n $ unitriangular矩阵的组。也就是说，$ u_n $由上三角$ n \ times n $矩阵组成，每个对角线条目等于$ 1 $。众所周知，$ u_n $的所有不可约的复杂字符的学位都是$ q $的功率。 Lehrer猜想是，$ u_n $ of度量$ q^e $的不可约字符的数量是$ q $的整数多项式，具体仅取决于$ e $和$ n $。我们表明，存在递归（对于$ n $）公式，当$ e $是$ 1、2 $和$ 3 $之一时，此数字可以满足该数字，从而表明在这种情况下，猜想是正确的。
Let $U_n$ denote the group of upper $n \times n$ unitriangular matrices over a fixed finite field $\mathbb{F}$ of order $q$. That is, $U_n$ consists of upper triangular $n \times n$ matrices having every diagonal entry equal to $1$. It is known that the degrees of all irreducible complex characters of $U_n$ are powers of $q$. It was conjectured by Lehrer that the number of irreducible characters of $U_n$ of degree $q^e$ is an integer polynomial in $q$ depending only on $e$ and $n$. We show that there exist recursive (for $n$) formulas that this number satisfies when $e$ is one of $1, 2$ and $3$, and thus show that the conjecture is true in those cases.
