Questions tagged [character-theory]
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32 questions
17
votes
1
answer
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What is the significance of the height of a character?
Let $G$ be a finite group, and fix a prime $p$ which divides $|G|$.
The ordinary (complex) characters $\text{Irr}(G)$ can be partitioned into what are called $p$-blocks, and to each block $B$ can be ...
- 2,413
16
votes
1
answer
612
views
When is the ring of integers of a character field the ‘character ring’?
Let $G$ be a finite group with an irreducible complex character $\chi$.
Let $\mathbb Q(\chi)$ denote the field extension of the rationals generated by the values of $\chi(g)$ for $g \in G$.
A theorem ...
- 333
0
votes
1
answer
209
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Complex roots of subgroup-representation polynomials
For a finite group $G$, define
$$\zeta(G) := \frac{1}{|G|} \sum_{\chi \in \text{Irr}(G)} \chi(1)^3$$
where the sum is over all irreducible complex characters and $\chi(1)$ is the degree.
Define the ...
6
votes
1
answer
151
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Real Schur indices for spin characters of symmetric groups
Let $\widetilde{S}_n^\pm,\widetilde{A}_n$ denote the double covers of the symmetric and alternating groups for $n\geq 4$. I would like to know the Schur indices over the reals (or equivalently, the ...
- 91
1
vote
0
answers
120
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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 $S_4$.
- 11
4
votes
0
answers
194
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 ...
- 2,413
2
votes
1
answer
306
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 ...
- 2,413
11
votes
1
answer
1k
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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 ...
- 2,413
4
votes
1
answer
500
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 ...
- 419
10
votes
1
answer
867
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 $\...
- 5,377
7
votes
2
answers
845
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 \...
- 5,389
11
votes
1
answer
312
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 ...
- 178
6
votes
0
answers
181
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 ...
- 81
1
vote
1
answer
212
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 ...
- 415
2
votes
0
answers
97
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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 ...
- 415