Shiv Gupta
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.
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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.
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.
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.
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.
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 $\mathrm{M}_{10}$.
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.
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。
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…
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].
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.
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.
An element $g$ of a finite group $G$ is said to be vanishing in $G$ if there exists an irreducible character $χ$ of $G$ such that $χ(g)=0$; in this case, $g$ is also called a zero of $G$. The aim of this paper is to obtain structural properties of a factorised group $G=AB$ when we impose some conditions on prime power order elements $g\in A\cup B$ which are (non-)vanishing in $G$.
Let $G$ be a finite nilpotent group, $χ$ and $ψ$ be irreducible complex characters of $G$ of prime degree. Assume that $χ(1)=p$. Then either the product $χψ$ is a multiple of an irreducible character or $χψ$ is the linear combination of at least $\frac{p+1}{2}$ distinct irreducible characters.
Let $G$ be a finite group and $p\in π(G)$, and let Irr$(G)$ be the set of all irreducible complex characters of $G$. Let $χ\in {\rm Irr}(G)$, we write ${\rm cod}(χ)=|G:{\rm ker} χ|/χ(1)$, and called it the codegree of the irreducible character $χ$. Let $N\unlhd G$, write ${\rm Irr}(G|N)=\{χ\in {\rm Irr}(G)~|~N\nsubseteq {\rm ker}χ\}$, and ${\rm cod}(G|N)=\{ {\rm cod}(χ) ~|~χ\in{\rm Irr}(G|N)\}.$ In this Ipaper, we prove that if $N\unlhd G$ and every member of ${\rm cod}(G|N')$ is not divisible by some fixed prime $p\in π(G)$, then $N$ has a normal $p$-complement and $N$ is solvable.
Let N be a normal subgroup of a finite group G and consider the set cd(G|N) of degrees of irreducible characters of G whose kernels do not contain N. A number of theorems are proved relating the set cd(G|N) to the structure of N. For example, if N is solvable, its derived length is bounded above by a function of |cd(G|N)|. Also, if |cd(G|N)| is at most 2, then N is solvable and its derived length is at most |cd(G|N)|. If G is solvable and |cd(G|N)| = 3, then the derived length of N is at most 3.
Sylow subgroups and the number of irreducible characters of degrees divisible by a prime $p$
本文研究了有限群中 Sylow p-子群的导出长度与可约特征标度数 divisible 由素数 p 的关系，证明了 p-块中缺陷群的导出长度与正高度特征标的数量之间的上界。
The weights for a finite group G with respect to a prime number p where introduced by Jon Alperin, in order to formulate his celebrated conjecture. In 1992, Everett Dade formulates a refinement of Alperin's conjecture involving ordinary irreducible characters - with their defect - and, in 2000, Geoffrey Robinson proves that the new conjecture holds for p-solvable groups. But this refinement is formulated in terms of a vanishing alternating sum, without giving any possible refinement for the weights. In this note we show that, in the case of the p-solvable finite groups, the method developed in…
A finite group $G$ is $normally ~monomial$ if all its irreducible characters are induced from linear characters of normal subgroups of $G$. In this paper, we determine all possible irreducible character degree sets of normally monomial 5-groups of maximal class. Moreover, we give an upper bound for the largest irreducible character degree of normally monomial $p$-groups of maximal class in terms of $p$.
A method to construct irreducible unitary representations of a hyperspecial compact subgroup of a reductive group over p-adic field with odd p is presented. Our method is based upon Cliffods theory and Weil representations over finite fields. It works under an assumption of the triviality of certain Schur multipliers defined for an algebraic group over a finite field. The assumption of the triviality has good evidences in the case of general linear groups and highly probable in regular cases in general. We will give several examples of classical groups where the Schur multipliers are actually …
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.
In this notes we construct and count all ordinary irreducible characters of Sylow $p$-subgroups of the Steinberg triality groups ${}^3D_4(p^{3m})$.
The irreducible character values of the spin wreath products of the symmetric group and an arbitrary finite group are completely determined.
Let $G$ be a finite group. We consider the set of the irreducible complex characters of $G$, namely $Irr(G)$, and the related degree set $cd(G)=\{χ(1) : χ\in Irr(G)\}$. Let $ρ(G)$ be the set of all primes which divide some character degree of $G$. In this paper we introduce the bipartite divisor graph for $cd(G)$ as an undirected bipartite graph with vertex set $ρ(G)\cup (cd(G)\setminus\{1\})$, such that an element $p$ of $ρ(G)$ is adjacent to an element $m$ of $cd(G)\setminus\{1\}$ if and only if $p$ divides $m$. We denote this graph simply by $B(G)$. Then by means of combinatorial properties…
We classify Sylow $p$-subgroups of finite groups whose principal $p$-blocks have precisely six ordinary irreducible characters.
In this paper we study finite p-solvable groups having irreducible complex characters chi in Irr(G) which take roots of unity values on the p-singular elements of G.
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。
In this note, we formulate an observation that "almost all" irreducible ordinary characters of finite groups of Lie type remain irreducible when restricted to the derived subgroups. To see this, key ingredients are some asymptotic results for conjugacy classes of finite groups of Lie type and strongly regular semisimple elements in dual groups.
Minimal-order groups with an irreducible character of degree $p$ or $p^2$
本研究刻画了具有次数为素数$p$或$p^2$的不可约复特征标的极小阶有限群的结构特征。
A complex irreducible character of a finite group G is said to be p-constant, for some prime p dividing the order of G, if it takes constant value at the set of p-singular elements of G. In this paper we classify irreducible p-constant characters for finite reflection groups, nilpotent groups and complete monomial groups. We also propose some conjectures about the structure of the groups admitting such characters.
We establish an irreducibility property for the characters of finite dimensional, irreducible representations of simple Lie algebras (or simple algebraic groups) over the complex numbers, i.e., that the characters of irreducible representations are irreducible after dividing out by (generalized) Weyl denominator type factors. For $SL(r)$ the irreducibility result is the following: let $λ=(a_1\geq a_2\geq ... a_{r-1}\geq 0)$ be the highest weight of an irreducible rational representation $V_λ$ of $SL(r)$. Assume that the integers $a_1+r-1, ~a_2+r-2,..., a_{r-1}+1$ are relatively prime. Then the…
We show that if the principal p-block of a finite group G contains exactly 5 irreducible ordinary characters, then a Sylow p-subgroup of G has order 5, 7 or is isomorphic to one of the non-abelian 2-groups of order 8.
Codegrees of Irreducible Characters of VZ and Camina $p$-Groups
通过计算VZ群和Camina $p$-群的不可约特征标的余次数，得到了阶数不超过$p^5$的$p$-群的特征标余次数集，其中$p$为奇素数。
The irreducible Brauer characters of the automorphism group of the Chevalley group F_4(2)
本研究利用计算方法推导谢瓦莱群F_4(2)自同构群的不可约Brauer特征标，结果仍存在一个参数和一处一致性问题待完善。
The object of this short note is to prove a theorem and present a conjecture for the number of even entries in the character table of the symmetric group.
In representation theory of finite groups an important role is played by irreducible characters of p-defect 0, for a prime p dividing the group order. These are exactly those vanishing at the p-singular elements. In this paper we generalize this notion investigating the irreducible characters that are constant at the p-singular elements. We determine all such characters of non-zero defect for alternating, symmetric and sporadic simple groups. We also classify the irreducible characters of quasi-simple groups of Lie type that are constant on the non-identity unipotent elements. In particular, w…
Relations between values and zeros of irreducible characters of symmetric groups
本文研究了对称群不可约特征标值与零点之间的关系，证明了某些多项式关系，并改进了通过检查少量值来识别不可约特征标的结果。
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.
Given a prime number $p$, every irreducible character $χ$ of a finite group $G$ determines a unique conjugacy class of $p$-subgroups of $G$ which we will call the anchors of $χ$. This invariant has been considered by L. Barker in the context of finite $p$-solvable groups. Besides proving some basic properties of these anchors, we investigate the relation to other $p$-groups which can be attached to irreducible characters, such as defect groups, vertices in the sense of J. A. Green and vertices in the sense of G. Navarro.
Common zeros of irreducible characters
本文研究有限群不可约特征的公共零点问题，引入公共零点图并证明了对称群特征公共零点存在性，且可解群与单群该图连通分支上界为3，最后应用于证明置换度量的不等价性。
This paper computes the irreducible characters of the alternating Hecke algebras, which are deformations of the group algebras of the alternating groups. More precisely, we compute the values of the irreducible characters of the semisimple alternating Hecke algebras on a set of elements indexed by minimal length conjugacy class representatives and we show that these character values determine the irreducible characters completely. As an application we determine a splitting field for the alternating Hecke algebras in the semisimple case.
On the irreducible character degrees of symmetric groups and their multiplicities
本文基于计算机实验，提出关于对称群最大不可约特征标次数及其重数的新猜想，证明当n≥21时S_n有至少八个同次数且限制到A_n不可约的特征标，并探讨GL_n(q)的类似问题。
Let $G_2(q)$ be the Chevalley group of type $G_2$ defined over a finite field with q=p^n elements, where p is a prime number and $n$ is a positive integer. In this paper, we determine when the restriction of an absolutely irreducible representation of $G$ in characteristic other than p to a maximal subgroup of $G_2(q)$ is still irreducible. Similar results are obtained for $^2B_2(q)$ and $^2G_2(q)$.
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