Edith Adan-Bante
We study the solvable groups $G$ that have an irreducible character $χ\in \Irr(G)$ such that $χ\barχ$ has at most two non-principal irreducible constituents.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
On the number of constituents of products of characters
本文针对有限p-群中两个忠实不可约特征标乘积的不可约组成成分数量猜想，构造并给出了一个反例。
Let $G$ be a finite solvable group and $χ, ψ\in \Irr(G)$ be complex characters of $G$. Let $α$ be an irreducible constituent of the product $χψ$. We show that the derived length of $\Ker(α)/\Ker(χψ)$ is bounded by a linear function on the number of distinct irreducible constituents of $χψ$
Let G be a finite solvable group and $χ\in \Irr(G)$ be a faithful character. We show that the derived length of G is bounded by a linear function of the number of distinct irreducible constituents of $χ\barχ$. We also discuss other properties of the decomposition of $χ\barχ$ into its irreducible constituents.
Let $G$ be a finite $p$-group and $χ,ψ$ be irreducible characters of $G$. We study the character $χψ$ when $χψ$ has at most $p-1$ distinct irreducible constituents.
Let G be a finite p-group, where p is a prime number, and $χ$ and $ψ$ be faithful complex irreducible characters of G. We study the relation between the number $η(χ,ψ)$ of distinct irreducible constituents of the product $χψ$ and p.
We extend the Howlett-Isaacs theorem on the solvability of groups of central type taking into account actions by automorphisms. Then we study certain induced characters whose constituents have all the same degree.
Let $G$ be a finite $p$-group, where $p$ is an odd prime number, $H$ be a subgroup of $G$ and $θ\in \Irr(H)$ be an irreducible character of $H$. Assume also that $|G:H|=p^2$. Then the character $θ^G$ of $ G$ induced by $θ$ is either a multiple of an irreducible character of $G$, or has at least $\frac{p+1}{2}$ distinct irreducible constituents.
We show that a tensor product of irreducible, finite dimensional representations of a simple Lie algebra over a field of characteristic zero, determines the individual constituents uniquely. This is analogous to the uniqueness of prime factorisation of natural numbers.
We study the decomposition of certain reducible characters of classical groups as the sum of irreducible ones. Let ${\mathbf G}$ be an algebraic group of classical type with defining characteristic $p>0$, $μ$ a dominant weight and $W$ the Weyl group of ${\mathbf G}$. Let $G=G(q)$ be a finite classical group, where $q$ is a $p$-power. For a weight $μ$ of ${\mathbf G}$ the sum $s_μ$ of distinct weights $w(μ)$ with $w\in W$ viewed as a function on the semisimple elements of $G$ is known to be a generalized Brauer character of $G$ called an orbit character of $G$. We compute, for certain orbit cha…
We determine the irreducible 2-modular representations of the symplectic group $G=Sp_{2n}(2)$ whose restriction to every abelian subgroup has a trivial constituent. A similar result is obtained for maximal tori of $G$. There is significant information on the existence of eigenvalue 1 of elements of $G$ in a given irreducible representation of $G$.
Let $G$ be a finite simple group of Lie type, and let $π_G$ be the permutation representation of $G$ associated with the action of $G$ on itself by conjugation. We prove that every irreducible representation of $G$ is a constituent of $π_G$, unless $G=PSU_n(q)$ and $n$ is coprime to $2(q+1)$, where precisely one irreducible representation fails. Let St be the Steinberg representation of $G$. We prove that a complex irreducible representation of $G$ is a constituent of the tensor square $St\otimes St$, with the same exceptions as in the previous statement.
Let $H$ be a linear algebraic group whose connected component $G\neq 1$ is simple and $H/G$ is cyclic. We determine the irreducible projective representations $ϕ$ of $H$ such that $ϕ(G)$ is irreducible and $ϕ(h)$ has simple spectrum for some $h\in H$. The latter means that all irreducible constituents of the group $ϕ( \langle h\rangle)$ are of multiplicity 1. (Here $\langle h\rangle$ is the subgroup of $H$ generated by $h$.) This extends an earlier known result for $H=G$.
Brauer character degrees and nilpotent subgroups
该研究解决了陈与纳瓦罗提出的关于布饶尔特征标和幂零子群的问题，在G/N可解且H/N幂零的条件下，证明了H的不可约ℓ-布饶尔特征标可提升为G的对应特征标，且满足次数整除关系。
We restrict irreducible characters of alternating groups of degree divisible by $p$ to their Sylow $p$-subgroups and study the number of linear constituents.
We construct a natural bijection between odd-degree irreducible characters of S_n and linear characters of its Sylow 2-subgroup P_n. When n is a power of 2, we show that such a bijection is nicely induced by the restriction functor. We conclude with a characterization of the irreducible characters χof S_n such that the restriction of χto P_n has a unique linear constituent.
A character of a group is said to be super-monomial if every primitive character inducing it is linear. It is conjectured by Isaacs that every irreducible character of an odd $M$-group is super-monomial. We show that all non linear irreducible characters of lowest degree of an odd $M$-group is super-monomial and provide cases in which one can guarantee that certain irreducible characters of normal subgroups are super-monomial. Finally, we study groups having two irreducible monomial character degrees.
The irreducible representations $ϕ_n^1$ and $ϕ_n^2$ of the symplectic group $G_n=Sp_{2n}(P)$ over an algebraically closednfield $P$ of characteristic $p>2$ with highest weights $ω_{n-1}+\frac{p-3}{2}ω_n$ and $\frac{p-1}{2}ω_n$, respectively, are investigated. It is proved that the dimension of $ϕ_n^i$ ($i=1,2$) is equal to $(p^n+(-1)^i )/2$, all weight multiplicities of these representations are equal to $1$, their restrictions to the group $G_k$ naturally embedded into $G_n$ are completely reducible with irreducible constituents $ϕ_k^1$ and $ϕ_k^2$, and their restrictions to $Sp_{2n}(p)$ can …
The deep theory of approximate subgroups establishes 3-step product growth for subsets of finite simple groups $G$ of Lie type of bounded rank. In this paper we obtain 2-step growth results for representations of such groups $G$ (including those of unbounded rank), where products of subsets are replaced by tensor products of representations. Let $G$ be a finite simple group of Lie type and $χ$ a character of $G$. Let $|χ|$ denote the sum of the squares of the degrees of all (distinct) irreducible characters of $G$ which are constituents of $χ$. We show that for all $δ>0$ there exists $ε>0$, in…
We determine the irreducible constituents of the Steinberg character of an orthogonal group over a finite field restricted to the orthogonal group of one less dimension
We classify the finite groups $G$ which satisfies the condition that every complex irreducible character,whose degree's square doesn't divide the index of its kernel in $G$, lies in the same Galois conjugacy class.
Let p be a prime number. Let G be a finite p-group and $χ\in Irr(G)$. Denote by $\barχ \in Irr(G)$ the complex conjugate of $χ$. Assume that $χ(1)=p^n$. We show that the number of distinct irreducible constituents of the product $χ\barχ$ is at least $2n(p-1)+1$.
On Frobenius graphs of diameter 3 for finite groups
本研究针对有限群的Frobenius图，探究其直径为3的成立条件，该图用于记录有限群不可约特征标限制到子群上的分量。
In this work, we classify all finite groups such that for every field extension F of \mathbb{Q}, F is the field of values of at most 3 irreducible characters.
In this paper we determine the ordinary irreducible characters of the five-dimensional full linear group over a Galois field of q elements. We use the techniques developed by J. A. Green.
Let G be a finite group and N be a non-trivial normal subgroup of G, such that the average character degree of irreducible characters in Irr(G|N) is less than or equal to 16=5. Then we prove that N is solvable. Also, we prove the solvability of G, by assuming that the average character degree of irreducible characters in Irr(G|N) is strictly less than 16=5. We show that the bounds are sharp.
For an irreducible character $χ$ of a finite group $G$, the codegree of $χ$ is defined as $|G:\ker(χ)|/χ(1)$. In this paper, we determine finite nonsolvable groups with exactly three nonlinear irreducible character codegrees, and they are $\mathrm{L}_2(2^f)$ for $f\ge 2$, $\mathrm{PGL}_2(q)$ for odd $q\ge 5$ or $\mathrm{M}_{10}$.
Let $U_n(q)$ denote the upper triangular group of degree $n$ over the finite field $\F_q$ with $q$ elements. It is known that irreducible constituents of supercharacters partition the set of all irreducible characters $\Irr(U_n(q)).$ In this paper we present a correspondence between supercharacters and pattern subgroups of the form $U_k(q)\cap {}^wU_k(q)$ where $w$ is a monomial matrix in $GL_k(q)$ for some $k<n.$
We consider groups where the centers of the irreducible characters form a chain. We obtain two alternate characterizations of these groups, and we obtain some information regarding the structure of these groups. Using our results, we are able to classify those groups where the kernels of the irreducible characters form a chain. We show that a result of Nenciu regarding nested GVZ groups is really a result about nested groups. We obtain an alternate proof of a theorem of Isaacs regarding the existence of $p$-groups with a given set of irreducible character degrees.
Suppose that $B$ is a Brauer $p$-block with defect group $D$. If $B$ exactly contains 4 irreducible characters, then we show that $D$ has order 4 or 5, assuming the Alperin--McKay conjecture.
We prove a p'-version of a classical theorem of Broline and Garrison. As a consequence, we obtain results on p-parts of character codegrees.
Isaacs has defined a character to be super monomial if every primitive character inducing it is linear. Isaacs has conjectured that if $G$ is an $M$-group with odd order, then every irreducible character is super monomial. We prove that the conjecture is true if $G$ is an $M$-group of odd order where every irreducible character is a $\{p \}$-lift for some prime $p$. We say that a group where irreducible character is super monomial is a super $M$-group. We use our results to find an example of a super $M$-group that has a subgroup that is not a super $M$-group.
A well-known theorem of Burnside says that if $ρ$ is a faithful representation of a finite group $G$ over a field of characteristic $0$, then every irreducible representation of $G$ appears as a constituent of a tensor power of $ρ$. In 1962, R. Steinberg gave a module theoretic proof that simultaneously removed the constraint on the characteristic, and allowed the group to be replaced by a monoid. Brauer subsequently simplified Burnside's proof and, moreover, showed that if the character of $ρ$ takes on $r$ distinct values, then the first $r$ tensor powers of $ρ$ already contain amongst them a…
p-Groups in which kernels of the non-linear irreducible characters are of equal order
本文研究了素数幂阶有限群中非线性不可约特征标核阶相等的特征，探讨了此类群的结构与性质。
The Baby Monster is the largest group with at most $2$ irreducible characters with the same degree
本文分类了不可约特征标次数重数至多2的有限群，证明Baby Monster是这类群中最大的群。
Motivated by the study of an Hecke action on iterated Shimura integrals undertaken in [H], in this appendix to [H] we prove that, for any prime $p \geq 5$ and for any integer $n \geq 1$, every complex irreducible representation of $G=\mathrm{SL}_2(\mathbb{Z}/p^n \mathbb{Z})$ that are trivial on $\mathbf{Z}(G)$ appears as an irreducible constituent of the conjugation representation of $G$.
In this paper we consider finite groups G satisfying the following condition: G has two columns in its character table which differ by exactly one entry. It turns out that such groups exist and they are exactly the finite groups with a non-trivial intersection of the kernels of all but one irreducible characters or, equivalently, finite groups with an irreducible character vanishing on all but two conjugacy classes. We investigate such groups and in particular we characterize their subclass, which properly contains all finite groups with non-linear characters of distinct degrees, which were ch…
Characterizing normal Hall subgroups by character degrees, II
本文证明了有限单经典群中真非平凡 Hall 子群的特征标次数性质，从而结合已有约化，给出了任意有限群中 Hall 子群正规性的充要条件：其诱导表示的不可约成分次数均为 $\pi'$ 数。
We propose upper bounds for the number of modular constituents of the restriction modulo $p$ of a complex irreducible character of a finite group, and for its decomposition numbers, in certain cases.
Let $G$ be a finite group and $d$ the degree of a complex irreducible character of $G$, then write $|G|=d(d+e)$ where $e$ is a nonnegative integer. We prove that $|G|\leq e^4-e^3$ whenever $e>1$. This bound is best possible and improves on several earlier related results.
We classify the finite groups whose non-linear irreducible characters that are not conjugate under the natural Galois action have distinct degrees, therefore extending the results in Berkovich et al. [Proc. Amer. Math. Soc. {\bf 115} (1992), 955-959] and Dolfi et al. [Israel J. Math. {\bf 198} (2013), 283-331].
A converse for a theorem of Gallagher
本文证明了有限群特征标理论中 Gallagher 定理的逆命题，并进一步证明了其 Brauer 版本的局部逆命题，以及该定理对 Isaacs π-部分特征标的类似结果及其局部逆命题。
Let $G$ be a finite group and let $\rm{Irr}(G)$ be the set of all irreducible complex characters of $G$. For a character $χ\in \rm{Irr}(G)$, the number $\rm{cod}(χ):=|G:\rm{ker}χ|/χ(1)$ is called the co-degree of $χ$. The set of co-degrees of all irreducible characters of $G$ is denoted by $\rm{cod}(G)$. In this paper, we show that for a non-trivial finite group $G$, $|\rm{Irr}(G)|=|\rm{cod}(G)|$ if and only if $G$ is isomorphic to the cyclic group $\mathbb{Z}_2$ or the symmetric group $S_3$.
In this paper, we prove the existence of a relation between the prime divisors of the order of a Sylow normalizer and the degree of characters having values in some small cyclotomic fields. This relation is stronger when the group is solvable.
Finite groups in which every irreducible character has either $p'$-degree or $p'$-codegree
本文针对素数$p$，定义了有限群不可约复特征标的余次数，给出了每个不可约复特征标要么具有$p'$-次数要么具有$p'$-余次数的有限群的分类。
On the degrees of irreducible characters fixed by some field automorphism in finite groups
本文证明了 Ito-Michler 定理的一个变体，研究了在阶为 $p$ 的 Galois 自同构 $σ$ 保持不变的不可约特征标次数不被素数 $p$ 整除时有限群的性质。
Finite groups with exactly two nonlinear irreducible $p$-Brauer characters
本文分类了恰有两个非线性不可约 $p$-Brauer 特征的有限群，并指出阶不被 $p$ 整除的情形已由 Pálfy 于 1981 年解决。
In this paper, we construct a group with three real irreducible characters whose Sylow 2-subgroup is an iterated central extension of a Suzuki 2-group. This answers a question raised by Moreto and Navarro who asked whether such a group exists.
We study the action of a finite group on the Riemann-Roch space of certain divisors on a curve. If $G$ is a finite subgroup of the automorphism group of a projective curve $X$ over an algebraically closed field and $D$ is a divisor on $X$ left stable by $G$ then we show the irreducible constituents of the natural representation of $G$ on the Riemann-Roch space $L(D)=L_X(D)$ are of dimension $\leq d$, where $d$ is the size of the smallest $G$-orbit acting on $X$. We give an example to show that this is, in general, sharp (i.e., that dimension $d$ irreducible constituents can occur). Connections…
On permutation characters of finite group
本文研究了有限群G的排列特征标，证明当G可解时，所有G关于M的排列特征标均为单幂特征标，解决了钱和杨提出的问题。
Let $p$ be an odd prime and let $n$ be a natural number. In this article we determine the irreducible constituents of the permutation module induced by the action of the symmetric group $\mathfrak{S}_n$ on the cosets of a Sylow $p$-subgroup $P_n$. As a consequence, we determine the number of irreducible representations of the corresponding Hecke algebra $\mathcal{H}(\mathfrak{S}_n, P_n, 1_{P_n})$.
The branching theorem expresses irreducible character values for the symmetric group $S_n$ in terms of those for $S_{n-1}$, but it gives the values only at elements of $S_n$ having a fixed point. We extend the theorem by providing a recursion formula that handles the remaining cases. It expresses these character values in terms of values for $S_{n-1}$ together with values for $S_n$ that are already known in the recursive process. This provides an alternative to the Murnaghan-Nakayama formula.
We are interested in determining the bound of the average of the degrees of the irreducible characters whose degrees are not divisible by some prime $p$ that guarantees a finite group $G$ of odd order is $p$-nilpotent. We find a bound that depends on the prime $p$. If we further restrict our average by fixing a subfield $k$ of the complex numbers and then compute the average of the degrees of the irreducible characters whose degrees are not divisible by $p$ and have values in $k$, then we will see that we obtain a bound that depends on both $p$ and $k$. Moreover, we find examples that make tho…
On the number of primitive and quasi-primitive irreducible characters in finite groups
本文证明有限群 $G$ 的本原与拟本原不可约特征标数量均为指数 $|G:G'|$ 的倍数。
On the degrees of irreducible characters fixed by some field automorphism, p-solvable groups
本文将实值不可约特征标奇次数蕴含正规Sylow 2-子群的结论推广到p-可解群，证明若p不整除被p阶域自同构固定的不可约特征标次数，则该群具有正规Sylow p-子群。
Let $N$ be a normal subgroup of a finite group $G$. In this paper, we consider the elements $g$ of $N$ such that $χ(g)\neq 0$ for all irreducible characters $χ$ of $G$. Such an element is said to be non-vanishing in $G$. Let $p$ be a prime. If all $p$-elements of $N$ satisfy the previous property, then we prove that $N$ has a normal Sylow $p$-subgroup. As a consequence, we also study certain arithmetical properties of the $G$-conjugacy class sizes of the elements of $N$ which are zeros of some irreducible character of $G$. In particular, if $N=G$, then new contributions are obtained.
Let G be a finite p-group, for some prime p, and $ψ, θ\in \Irr(G)$ be irreducible complex characters of G. It has been proved that if, in addition, $ψ,θ$ are faithful characters, then the product $ψθ$ is a multiple of an irreducible or it is the nontrivial linear combination of at least $\frac{p+1}{2}$ distinct irreducible characters of G. We show that if we do not require the characters to be faithful, then given any integer k>0, we can always find a p-group G and irreducible characters $Ψ$ and $Θ$ such that $ΨΘ$ is the nontrivial combination of exactly k distinct irreducible characters. We d…
On the Sum of Squares of Irreducible Brauer Character Degrees in Blocks
本文研究了霍姆-威尔姆斯局部猜想的弱化版本，假设缺陷群为阿贝尔群时将问题简化为准单群，并在p=2时给出了完整证明。
In this paper, we determine new characterizations of nested and nested GVZ-groups, including character-free characterizations, but we additionally show that nested groups and nested GVZ-groups can be defined in terms of the existence of certain normal series.
We define and study supercharacters of the classical finite unipotent groups of symplectic and orthogonal types (over any finite field of odd characteristic). We show how supercharacters for groups of those types can be obtained by restricting the supercharacter theory of the finite unitriangular group, and prove that supercharacters are orthogonal and provide a partition of the set of all irreducible characters. We also describe all irreducible characters of maximum degree in terms of the root system, and show how they can be obtained as constituents of particular supercharacters.
We prove a broad generalization of a theorem of W. Burnside on real characters using permutation characters. Under a necessary hypothesis, We can give some control on multiplicities (a result that needs the Classification of Finite Simple Groups). Along the way, we also give a new characterization of the 2-closed finite groups using odd-order real elements of the group. All this can be seen as a contribution to Brauer's Problem 11. We also obtain similar results for 2-Brauer characters. We also classify finite primitive permutation groups in which every real element has a fixed point.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。