Questions tagged [character-theory]
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27
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How much of the ATLAS of finite groups is independently checked and/or computer verified?
In a recent talk Finite groups, yesterday and today Serre made some comments about proofs that rely on the classification of finite simple groups (CFSG) and on the ATLAS of Finite Groups. Namely, he ...
- 33.2k
17
votes
1
answer
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Number of solutions to equations in finite groups
Suppose $G$ is a finite group and that $E$ is an equation of the form $x_1 x_2 ... x_n = e$, where each $x_i$ is in the set of symbols $\{x, y, x^{-1}, y^{-1}\}$.
Is it always true that the number ...
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1
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Does $\chi(1)^2=|G:Z(G)|$ for irreducible character of a finite group $G$ imply $G$ is solvable?
In "Character Theory of Finite Groups" I.M. Isaacs mention the following conjecture:
It is only possible in a solvable group $G$ to have $\chi(1)^2=|G:Z(G)|$ with $\chi \in$ Irr$(G)$.
Is this ...
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Can the numerator in Weyl's character formula be written as a determinant?
I paraphrase part of the wikipedia article on the Weyl character formula: Weyl character formula.
If $\pi$ is an irreducible finite-dimensional representation of a complex semisimple Lie algebra $\...
- 4,769
9
votes
1
answer
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Representations of finite groups over the "field with one element"
Have there been any attempts to extend the "F_un" analogy to the representation theory of finite groups?
If I might be allowed some speculation:
If combinatorics can be regarded as analagous ...
- 917
8
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3
answers
1k
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Beyond Brauer's theorem
Brauer's classic theorem states that any character of a finite group can be expressed as a linear combination with integer coefficients of characters induced by linear characters of p-elementary ...
- 17.4k
7
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2
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Proofs of a character identity?
Let $G$ be a finite group, $g \geq 0, k\geq 1 $ integers, and $(C_1,...,C_k)$ a tuple of conjugacy classes of $G$. I am interested in proofs of the following identity
$$
\sum_{(c_1,...,c_k) \in C_1 \...
- 5,309
6
votes
2
answers
350
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Subgroups from which all class functions extend to class functions on the ambient group
Until someone suggests better terminology, let me call a subgroup H of a finite group G segregated if every class function on H can be extended to a class function on G. Equivalently, H should have ...
- 25.3k
6
votes
1
answer
330
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A constant associated to the character table of a finite group
Following on from some of myprevious MO questions on finite group theory...
$\newcommand{\Irr}{\operatorname{Irr}}\newcommand{\Conj}{\operatorname{Conj}}\newcommand{\AMZL}{{\rm AM}_{\rm Z}}$
Let $G$ ...
- 25.3k
6
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0
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112
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Schur indices for 2-groups
I am looking for any results on Schur indices over $\mathbb{Q}$ for 2-groups. By a theorem of Roquette (corollary 10.14 in Isaacs) these numbers are at most 2. I am interested in 2-groups for which ...
- 81
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Information about permutation character from local action
Let $G$ be a finite permutation group acting transitively, but not regularly, on a set $V$. Let $H$ be the stabilizer of some point $v\in V$, and suppose that $H$ acts 2-transitively on one of its (...
- 11.1k
4
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147
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New characters from old
(All groups in the following discussion are assumed to be finite.)
Character induction is an operation that produces a character of a group given a character of a subgroup. I'm aware that there are ...
- 917
3
votes
2
answers
653
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reference request for character theory of p-extraspecial groups
In a recent preprint 1302.1929 my co-authors and I make use of some results about the character theory of extraspecial groups. These were largely gleaned from haphazard Googling, comments made by ...
- 25.3k
3
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1
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Bounding the derived length of a solvable group given the degrees of the irreducible monomial characters
Much is known about the derived length of a solvable group given the degrees and cardinality of the set of degrees of the irreducible characters. Martin Isaacs and Donald Passman pretty much started ...
- 395
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0
answers
52
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Index of subgroup generated by characters induced from $p$-elementary subgroups in the ring of virtual characters
I posted this over on MSE, but received absolutely no love. So maybe I’ll have better luck here. It seems like a relatively easy group theory question that I’m just not seeing! It’s on the essential ...
- 131
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2
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Character theory of representations of infinite groups
I saw that "Over an algebraically closed field of characteristic 0, semisimple representations are isomorphic if and only if they have the same character" in the Wikipedia page , which does not ...
- 159
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1
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Relation between spectra of a Cayley graph of a group and irreducible characters of that group
I know the following fact:
If $G$ is an abelian group and $S\subset G$ be a subset of G such that $1\notin G$ and $S=S^{-1}$ and we draw an edge between $g$ and $h$ if and only if $hg^{-1}\in S$,then ...
- 131
2
votes
1
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An arithmetic problem involving a system of equations
Fix a positive integer $r$. Describe the solutions to the system of equations given by:
$$\begin{equation}\sum_{1\leq i\leq r}X_i^2\equiv0\pmod{X_k}(1\leq k\leq r)\end{equation}$$
Example: In the case ...
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0
answers
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Properties of extendable irreducible characters to a normal Sylow subgroup
Let $G$ be a finite group with a nilpotent commutator subgroup, and denote by $mcd(G)$ the set of degrees of the irreducible monomial characters. Suppose $|mcd(g)|=2$. Furthermore, we call a monomial ...
- 395
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The largest number of irreducible characters of the same degree in a finite group
Dear all,
For a finite group $G$, let $m(G)$ denote the largest number of irreducible characters of the same degree of $G$. You can say that $m(G)$ is the largest multiplicity of character degrees of ...
- 377
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1
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402
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Are there any groups $G$ with the property $(*_d)$?
Let $G$ be a finite group of even order which has only one non-principal irreducible character $\chi$ of degree $d$, $d\in \mathbb{N}$, with the following property (we name it $(*_d)$):
$(*_d)$: ...
- 79
1
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1
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Even Counterexample to Statement About the Non Existence of Certain Groups with Two Irreducible Monomial Character Degrees
Let $\textrm{cd}(G)=\lbrace \chi(1)\,|\, \chi\in\textrm{Irr}(G)\rbrace$ denote the set of character degrees of a finite group $G$. Similarly, denote by $\textrm{mcd}(G)$ the set of monomial character ...
- 395
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1
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111
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Study of the subgroups of which a non-linear monomial character is induced from
Let G be a monomial group, and let H_1,...,H_r be the subgroups of G where there exists a linear character that induces to an irreducible character of G. How much is known about these subgroups? For ...
- 395
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0
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56
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Irreducible characters of a semi-direct product with a p-group
Suppose G is a semi-direct product of P with H where P is a (non-abelian) p-group and G is solvable. I wonder what can be said about the irreducible characters of G given information about the ...
- 395
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If the kernel of an irreducible character contains the derived subgroup - is it then linear? [closed]
It is of course true that any linear character of a group contains the commutator subgroup. But is the converse also true? If not - do you know of a counterexample?
- 395
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We know $A_5$ as a non-CI-group. Now, is $A_5$ a BI-group?
We call a group satisfying the following property for all $\nu \in cd(G)$ (Irreducible character degrees of $G$) a BI-group (Babai Invariant group)
Let $G$ be a finite group, let $\Gamma=Cay(G,S)...
- 79
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Do you know any clear classification of groups in which there would exist a unique non-linear character of a given degree?
According to
Lev Kazarin, On Thompson’s Theorem, Journal of Algebra 220, 574–590 (1999)
we know that:
[Corollary 5.3]:Let $$cd(G)=\{\chi(1)|\chi\in Irr(G)\}=\{1,f_1,\dots,f_n,d\}, \;\;n\gt0,$$
...
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