计算上单位三角群中小度数的特征标

三角矩阵与单位三角矩阵的分支规则与交换概率

Branching rules and commuting probabilities for Triangular and Unitriangular matrices

本文通过计算分支规则，枚举了奇特征有限域上三角群与单位三角群中小规模情形的同时共轭类，并利用分支矩阵显式求出了相应的交换概率。

Exotic characters of unitriangular matrix groups

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 th…

The pro-$p$ group of upper unitriangular matrices

We study the pro-$p$ group $G$ whose finite quotients give the prototypical Sylow $p$-subgroup of the general linear groups over a finite field of prime characteristic $p$. In this article, we extend the known results on the subgroup structure of $G$. In particular, we give an explicit embedding of the Nottingham group as a subgroup and show that it is selfnormalising. Holubowski (\cite{holub1,holub0,holub2}) studies a free product $C_p*C_p$ as a (discrete) subgroup of $G$ and we prove that its closure is selfnormalising of infinite index in the subgroup of $2$-periodic elements of $G$. We als…

有限辛群和正交群的Sylow p-子群的超特征标

Supercharacters of the Sylow p-subgroups of the finite symplectic and orthogonal groups

本文定义并研究了（在任何奇特征的有限域上）辛型和正交型的经典有限幂幺群的超特征标。我们展示了如何超字符群的这些类型可以通过限制超字符理论的有限unitriangular群，并证明了超字符是正交的，并提供了一个分区的所有不可约字符集。我们还描述了所有的不可约字符的最大程度的根系，并显示它们如何可以获得的成分，特别是超字符。

Bounding Harish-Chandra series

We use the progenerator constructed in our previous paper to give a necessary condition for a simple module of a finite reductive group to be cuspidal, or more generally to obtain information on which Harish-Chandra series it can lie in. As a first application we show the irreducibility of the smallest unipotent character in any Harish-Chandra series. Secondly, we determine a unitriangular approximation to part of the unipotent decomposition matrix of finite orthogonal groups and prove a gap result on certain Brauer character degrees.

An equivariant bijection between irreducible Brauer characters and weights for Sp(2n, q)

The longstanding Alperin weight conjecture and its blockwise version have been reduced to simple groups recently by Navarro, Tiep, Spaeth and Koshitani. Thus, to prove this conjecture, it suffices to verify the corresponding inductive condition for all finite simple groups. The first is to establish an equivariant bijection between irreducible Brauer characters and weights for the universal covering groups of simple groups. Assume q is a power of some odd prime p. We first prove the blockwise Alperin weight conjecture for Sp2n(q) and odd non-defining characteristics. If the decomposition matri…

Supercharacters of discrete algebra groups

The concept of a supercharacter theory of a finite group was introduced by Diaconis and Isaacs as an alternative to the usual irreducible character theory, and exemplified with a particular construction in the case of finite algebra groups. We extend this construction to arbitrary countable discrete algebra groups, where superclasses and indecomposable supercharacters play the role of conjugacy classes and indecomposable characters, respectively. Our construction can be understood as a cruder version of Kirillov's orbit method and a generalisation of Diaconis and Isaacs construction for finite…

On the decomposition numbers of $\mathrm{SO}_8^+(2^f)$

Let $q=2^f$, and let $G=\mathrm{SO}_8^+(q)$ and $U$ be a Sylow $2$-subgroup of $G$. We first describe the fusion of the conjugacy classes of $U$ in $G$. We then use this information to prove the unitriangularity of the $\ell$-decomposition matrices of $G$ for all $\ell \ne 2$ by inducing certain irreducible characters of $U$ to $G$; the characters of $U$ of degree $q^3/2$ play here a major role. We then determine the $\ell$-decomposition matrix of $G$ in the case $\ell \mid q+1$, when $\ell \ge 5$ and $(q+1)_\ell>5$, up to two non-negative indeterminates in one column.

Counting lifts of Brauer characters

In this paper we examine the behavior of lifts of Brauer characters in p-solvable groups where p is an odd prime. In the main result, we show that if ϕ\in IBrp(G) is a Brauer character of a solvable group such that ϕhas an abelian vertex subgroup Q, then the number of lifts of ϕin Irr(G) is at most |Q|. In order to accomplish this, we develop several results about lifts of Brauer characters in p-solvable groups that were previously only known to be true in the case of groups of odd order.

Counting conjugacy classes of cyclic subgroups for fusion systems

We give another proof of an observation of Thévenaz \cite{T1989} and present a fusion system version of it. Namely, for a saturated fusion system $\CF$ on a finite $p$-group $S$, we show that the number of the $\CF$-conjugacy classes of cyclic subgroups of $S$ is equal to the rank of certain square matrices of numbers of orbits, coming from characteristic bisets, the characteristic idempotent and finite groups realizing the fusion system $\CF$ as in our previous work \cite{P2010}.

Counting characters in blocks of solvable groups with abelian defect group

If $G$ is a solvable group and $p$ is a prime, then the Fong-Swan theorem shows that given any irreducible Brauer character $ϕ$ of $G$, there exists a character $χ\in \irrg$ such that $χ^o = ϕ$, where $^o$ denotes the restriction of $χ$ to the $p$-regular elements of $G$. We say that $χ$ is a {\it{lift}} of $ϕ$ in this case. It is known that if $ϕ$ is in a block with abelian defect group $D$, then the number of lifts of $ϕ$ is bounded above by $|D|$. In this paper we give a necessary and sufficient condition for this bound to be achieved, in terms of local information in a subgroup $V$ determi…

不可约Brauer特征标提升的计数

Counting lifts of irreducible Brauer characters

针对$p$-可解群中不可约Brauer特征标的提升计数问题，本文在顶点$Q$条件更弱的情况下，显式描述了具有给定顶点对$(Q,δ)$的提升集合，推广了2011年Cossey等人关于$Q$正规时提升数量至多为$|Q:Q'|$的结果。

Counting real characters after Gallagher

Let $G$ be a finite group, let $N$ be a normal subgroup of $G$ and let $θ$ be an irreducible character of $N$. We count the real irreducible characters of $G$ lying over $θ$

计算线性群动作中的字符数

Counting characters in linear group actions

设$G$是有限群，$V$是有限$G$ -模.我们提出了上界的基数的某些子集的$\Irr(GV)$，如这些$χ\in\Irr(GV)$的集合，这样，对于一个固定的$v\in V$，限制的$χ$到$<v>$是不是一个倍数的常规字符的$<v>$。这些结果对于求解非互质$k(GV)$问题可能是有用的.

Counting lattices in simple Lie groups: the positive characteristic case

In this article we prove a conjecture of A. Lubotzky: let G = G_0(K), where K is a local field of characteristic p>5, G_0 is a simply connected, absolutely almost simple K-group of K-rank at least 2. We give the rate of growth of r_x(G) :={H< G| H a lattice in G; vol(G/H)<x}/~, where H_1\simH_2 if and only if there is an abstract automorphism t of G such that H_2 =t(H_1). We also study the rate of subgroup growth s_x(H) of any lattice H in G. As a result we show that these two functions have the same rate of growth which proves Lubotzky's conjecture. Along the way, we also study the rate of gr…

利用交替结构的循环计数与特征期望

Cycle Counting and Character Expectations Using Alternating Structures

本研究推广堆叠与双滑结构为交替结构，强化w-循环定理及相关结果，证明一般单词存在交替双滑结构，解决三个猜想（对一般单词成立）并为前两个找到反例。