Counting lifts of Brauer characters

Lifts and vertex pairs in solvable groups

Suppose $G$ is a $p$-solvable group, where $p$ is odd. We explore the connection between lifts of Brauer characters of $G$ and certain local objects in $G$, called vertex pairs. We show that if $χ$ is a lift, then the vertex pairs of $χ$ form a single conjugacy class. We use this to prove a sufficient condition for a given pair to be a vertex pair of a lift and to study the behavior of lifts with respect to normal subgroups.

Navarro vertices and lifts in solvable groups

Let $Q$ be a $p$-subgroup of a finite $p$-solvable group $G$, where $p$ is a prime, and suppose that $δ$ is a linear character of $Q$ with the property that $δ(x)=δ(y)$ whenever $x,y\in Q$ are conjugate in $G$. In this situation, we show that restriction to $p$-regular elements defines a canonical bijection from the set of those irreducible ordinary characters of $G$ with Navarro vertex $(Q,δ)$ onto the set of irreducible Brauer characters of $G$ with Green vertex $Q$. Also, we use this correspondence to examine the behavior of lifts of Brauer characters with respect to normal subgroups.

Dade–Glauberman–Nagao对应之上不可约Brauer特征标的等变双射

An equivariant bijection of irreducible Brauer characters above the Dade-Glauberman-Nagao correspondence

本文利用$\hH$-三元组框架，证明DGN对应之上存在保持顶点且对Galois与群自同构等变的不可约Brauer特征标双射，从而建立Galois Alperin权猜想的约化定理。

A solution to Brauer's Problem 14

It is well known that the number of real irreducible characters of a finite group G coincides with the number of real conjugacy classes of G. Richard Brauer has asked if the number of irreducible characters with Frobenius-Schur indicator 1 can also be expressed in group theoretical terms. We show that this can done by counting solutions of $g_1^2\ldots g_n^2=1$ with $g_1,\ldots,g_n\in G$.

Counting conjectures and $e$-local structures in finite reductive groups

We prove new results in generalized Harish-Chandra theory providing a description of the so-called Brauer--Lusztig blocks in terms of the information encoded in the $\ell$-adic cohomology of Deligne--Lusztig varieties. Then, we propose new conjectures for finite reductive groups by considering geometric analogues of the $\ell$-local structures that lie at the heart of the local-global counting conjectures. For large primes, our conjectures coincide with the counting conjectures thanks to a connection established by Broué, Fong and Srinivasan between $\ell$-structures and their geometric counte…

The Brauer indecomposability of Scott modules with wreathed $2$-group vertices

We give a sufficient condition for the $kG$-Scott module with vertex $P$ to remain indecomposable under taking the Brauer construction for any subgroup $Q$ of $P$ as $k[Q\,C_G(Q)]$-module, where $k$ is a field of characteristic $2$, and $P$ is a wreathed $2$-subgroup of a finite group $G$. This generalizes results for the cases where $P$ is abelian and some others. The motivation of this paper is that the Brauer indecomposability of a $p$-permutation bimodule ($p$ is a prime) is one of the key steps in order to obtain a splendid stable equivalence of Morita type by making use of the gluing met…

有限阿贝尔群的不可约表示计数

Counting Irreducible Representations of a Finite Abelian Group

针对素数幂阶域上与素数互素的有限阿贝尔群，本文利用Brauer特征标得到给定维数不可约表示数公式，结合指数素分解给出维数公式并提供计算不可约度数及重数的算法。

奇素数阶元素的Brauer-Fowler界

Brauer-Fowler Bounds for Elements of Odd Prime Order

本文研究奇素数阶元素的Brauer-Fowler界，证明Skresanov的对合相关结果的直接类似物对奇素数阶元素不成立，给出了有限单群阶数由中心化子内p阶元素数量及指数界定的结论。

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…

On Blocks with One Modular Character

Suppose that $B$ is a Brauer $p$-block of a finite group $G$ with a unique modular character $φ$. We prove that $φ$ is liftable to an ordinary character of $G$ (which moreover is $p$-rational for odd $p$). This confirms the basic set conjecture for these blocks.

单酉特征的一个局部-全局原理

A local-global principle for unipotent characters

本文针对类型A、B、C的单连通有限约化群，将Dade猜想和Späth特征三元组猜想适配到单酉特征，给出计数公式，并构造与特征三元组同构相容的参数化，提供局部-全局原理的几何版本。

Parity patterns associated with lifts of Hecke groups

Let $q$ be an odd prime, $m$ a positive integer, and let $\Ga_m(q)$ be the group generated by two elements $x$ and $y$ subject to the relations $x^{2m}=y^{qm}=1$ and $x^2=y^q$; that is, $\Ga_m(q)$ is the free product of two cyclic groups of orders $2m$ respectively $qm$, amalgamated along their subgroups of order $m$. Our main result determines the parity behaviour of the generalized subgroup numbers of $\Ga_m(q)$ which were defined in [T. W. Müller, Adv. in Math. 153 (2000), 118-154], and which count all the homomorphisms of index $n$ subgroups of $\Ga_m(q)$ into a given finite group $H$, in …