arXiv 1012.21922010-12-10math.RTmath.GR

单三角矩阵群的奇异特征标

Exotic characters of unitriangular matrix groups

Eric Marberg

备注：: 24 pages, 3 figures
发表信息：: J. Pure Appl. Algebra 216 (2012), 239-254
DOI：: 10.1016/j.jpaa.2011.06.003

摘要：

设$\UT_n(q)$表示有限域上具有基数$q$和素特征$p$的幂单$n\times n$上三角矩阵的单三角群.一段时间以来，人们已经知道，当$p$固定并且$n$足够大时，$\UT_n(q)$具有取分圆域$\QQ(ζ_p)$之外的值的“外来”不可约特征标。然而，迄今为止，这一事实的所有证明都是非建设性的，而且依赖于计算机。在前期工作中，我们定义了一个正交特征标族，它分解了任意代数群的超特征标。通过将这种构造应用于单三角群，我们能够手工导出取值于任意大分圆域的$\UT_n(q)$的一族特征标的显式描述。特别是，我们证明了，如果$r$是一个正整数幂的$p$和$n>6r$，那么$\UT_n(q)$有一个不可约字符的程度$q^{5r^2-2r}$它采取值以外的$\QQ(ζ_{pr})$。当$n>12$和$q$是任意的时，利用同样的技巧，我们还可以构造不属于$\UT_n(q)$的特征的显式Kirillov函数.

英文摘要：

Let $\UT_n(q)$ denote the unitriangular group of unipotent $n\times n$ upper triangular matrices over a finite field with cardinality $q$ and prime characteristic $p$. It has been known for some time that when $p$ is fixed and $n$ is sufficiently large, $\UT_n(q)$ has ``exotic'' irreducible characters taking values outside the cyclotomic field $\QQ(ζ_p)$. However, all proofs of this fact to date have been both non-constructive and computer dependent. In a preliminary work, we defined a family of orthogonal characters decomposing the supercharacters of an arbitrary algebra group. By applying this construction to the unitriangular group, we are able to derive by hand an explicit description of a family of characters of $\UT_n(q)$ taking values in arbitrarily large cyclotomic fields. In particular, we prove that if $r$ is a positive integer power of $p$ and $n>6r$, then $\UT_n(q)$ has an irreducible character of degree $q^{5r^2-2r}$ which takes values outside $\QQ(ζ_{pr})$. By the same techniques, we are also able to construct explicit Kirillov functions which fail to be characters of $\UT_n(q)$ when $n>12$ and $q$ is arbitrary.

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