Questions tagged [character-theory]

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How to know the character table of the twisted group algebra of the symmetric group $S_4$

Given the character table of its Schur cover group, is there a way to obtain the character table of twisted group algebra from that? I am particularly interested in the symmetric group $S_4$.
Wenxia Wu's user avatar
4 votes
0 answers
147 views

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 ...
semisimpleton's user avatar
2 votes
1 answer
250 views

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 ...
semisimpleton's user avatar
9 votes
1 answer
870 views

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 ...
semisimpleton's user avatar
3 votes
1 answer
271 views

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 ...
Soumyadip Sarkar's user avatar
10 votes
1 answer
690 views

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 $\...
Malkoun's user avatar
  • 4,981
7 votes
2 answers
735 views

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 \...
user101010's user avatar
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11 votes
1 answer
232 views

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 ...
Anton B's user avatar
  • 178
6 votes
0 answers
114 views

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 ...
John McHugh's user avatar
1 vote
1 answer
154 views

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 ...
Joakim Færgeman's user avatar
2 votes
0 answers
67 views

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 ...
Joakim Færgeman's user avatar
3 votes
1 answer
145 views

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 ...
Joakim Færgeman's user avatar
1 vote
0 answers
59 views

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 ...
Joakim Færgeman's user avatar
1 vote
1 answer
112 views

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 ...
Joakim Færgeman's user avatar
1 vote
0 answers
205 views

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?
Joakim Færgeman's user avatar
2 votes
2 answers
1k views

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 ...
Longma's user avatar
  • 169
3 votes
0 answers
52 views

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 ...
Nicholas Camacho's user avatar
1 vote
0 answers
111 views

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)...
M. Zallaghi's user avatar
8 votes
3 answers
1k views

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 ...
Myshkin's user avatar
  • 17.4k
53 votes
5 answers
5k views

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 ...
David Roberts's user avatar
  • 33.8k
17 votes
1 answer
757 views

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 ...
Paul Boddington's user avatar
6 votes
2 answers
353 views

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 ...
Yemon Choi's user avatar
  • 25.5k
1 vote
1 answer
402 views

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 $(*_d)$): $(*_d)$: ...
M. Zallaghi's user avatar
1 vote
0 answers
152 views

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 $$cd(G)=\{\chi(1)|\chi\in Irr(G)\}=\{1,f_1,\dots,f_n,d\}, \;\;n\gt0,$$ ...
M. Zallaghi's user avatar
6 votes
1 answer
333 views

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 $G$ ...
Yemon Choi's user avatar
  • 25.5k
3 votes
2 answers
668 views

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 ...
Yemon Choi's user avatar
  • 25.5k
2 votes
0 answers
167 views

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 ...
Uep's user avatar
  • 377
5 votes
0 answers
180 views

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 (...
Nick Gill's user avatar
  • 11.2k