All Questions
65 questions
4
votes
0
answers
180
views
Subgroups that conjugate-cover the ambient group
Let $G$ be a finite group, and suppose that a set of proper subgroups $H_1,\dotsc,H_n$ satisfy $G=\bigcup_{g\in G}\bigcup_{i=1}^nH_i^g$, where $H_i^g$ is the conjugate of $H_i$ by $g$. In this case, ...
2
votes
0
answers
97
views
Centralizer bound for irreducible representations of $\operatorname{SU}_n(\mathbb{C})$
Let $G$ be a finite group, $χ$ be the character of an irreducible representation $V$ of $G$, and $g ∈ G$. Then a classical bound on the trace of $g$ is given by: $|χ(g)|² ≤ |C_G(g)|$, where $C_G(g)$ ...
- 3,054
1
vote
0
answers
172
views
Isomorphism classes of finite $\mathbb{N}$-groups
Where can I find resources on isomorphism classes of finite $\mathbb{N}$-groups, i.e. groups acted on by the monoid $(\mathbb{N}, +)$?
I edited this question to be more focused on what I'm interested ...
- 63.9k
1
vote
0
answers
127
views
Irreducible projective representations of finite abelian groups
I want to know if there is a description of all irreducible complex projective representations of an arbitrary finite abelian group. I have seen this for particular cases such as those given here and ...
- 63.9k
1
vote
0
answers
110
views
Character table of $\mathrm{P\Gamma L}_2(q)$ with $q$ even
Let $q = 2^f$ for some integer $f\geqslant 3$. The character table of $\mathrm{SL}_2(q)\cong\mathrm{PSL}_2(q)$ can be deduced from the character table of $\mathrm{GL}_2(q)$ (see, for example, Exercise ...
- 379
35
votes
6
answers
5k
views
Character-free proof that Frobenius kernel is a normal subgroup?
The question is in the title, but here is some background/reminders:
A subgroup $H\neq\{1\}$ of a finite group $G$ is called a Frobenius complement if $H\cap H^g = \{1\}$ for all $g\in G\backslash H$....
- 63.9k
4
votes
1
answer
273
views
Wedderburn decomposition of special linear groups
$\DeclareMathOperator\SL{SL}\newcommand\card[1]{\lvert#1\rvert}$I want to study about Wedderburn decomposition of group algebra $k\SL(n,\mathbb{F}_p)$ where $k$ is either an algebraically closed field ...
- 13k
12
votes
2
answers
926
views
Finite groups with integral character table
The character table of a finite group will be called integral if all its entries are integers. There are $11$ such groups up to order $16$, namely $C_1$, $C_2$, $C_2^2$, $S_3$, $D_8$, $Q_8$, $C_2^3$, $...
- 44.4k
1
vote
0
answers
97
views
Minimizing distance over finite group action
Let $G$ be a finite group and $V$ a unitary irreducible rep’n of dimension $N$. Is there a fast (polynomial in $\log|G|$) algorithm to compute $\displaystyle \min_{g \in G}d(x,gy)=\max_{g \in G} Re\...
- 571
16
votes
3
answers
1k
views
Reference for representation theory of SL_2(Z/n)
There are many references for the representation theory (say over $\mathbf C$) of $\operatorname{SL}_2(\mathbf{F}_q)$ and $\operatorname{GL}_2(\mathbf{F}_q)$, for instance lecture 5 in Fulton--Harris &...
10
votes
7
answers
2k
views
Representations of products of symmetric groups
I'm writing a paper and want to cite some references to efficiently prove that over any field $k$ of characteristic zero, every irreducible representation of a product of symmetric groups, say
$$ S_{...
- 12.9k
3
votes
1
answer
305
views
What corresponds to the operation of taking traces in of the Fourier transformation on a finite group?
I have a question about the Fourier transfomation on a finite non-comutative group. I hope that it is a known fact in the Representation Theory but I cannot find it written explicitly in textbooks.
...
- 38.6k
3
votes
3
answers
356
views
Absolutely irreducible finite reflection/rotation groups
Suppose that $G$ is a finite irreducible reflection group with irreducible orthogonal representation $\rho: G\rightarrow \mathrm{O}(d)$, and let $\rho^+: G^+\rightarrow \mathrm{SO}(d)$ be its ...
- 11.2k
1
vote
0
answers
213
views
Is there any research on the action of a subgroup on the whole finite group by conjugation?
I want to know whether there are any research on the orbits of the action of a subgroup by conjugation on the whole group, when the group is finite. (Especially whole symmetric group.)
I'm especially ...
- 1,013
13
votes
3
answers
1k
views
Characterization of Frobenius complements
I have learned that Frobenius complements are characterized (among finite groups) by having a fixed point free complex representation.
That is, a finite group $G$ is a Frobenius complement if and only ...
- 118k
0
votes
1
answer
143
views
Faithful representation of group of order $p^4$
In the (xi) group of the classification of groups of order $p^4$ given by W.Burnside in his book, "Theory of groups of finite order". The group ($\mathbb{Z}_{p^{2}}\rtimes \mathbb{Z}_{p^{}}) ...
- 44.4k
2
votes
2
answers
250
views
In which books we can find structure information for finite simple groups and their Schur covering groups?
In which books we can find representations or character tables, Sylow 2-subgroups and conjugacy classes for finite simple groups and their Schur covering groups and properties for Schur multiplier of ...
- 271
5
votes
1
answer
152
views
How to find a finite splitting field $K$ for $G$ such that every indecomposable $KG$-module is absolutely indecomposable
Let $G$ be a finite group and let $k$ be a finite field with char$(k)=p$ such that $p\mid |G|$.
If $k$ is a splitting field for $G$, then, no matter which splitting field we take, after extending ...
- 63.9k
6
votes
0
answers
122
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 ...
- 63.9k
21
votes
2
answers
2k
views
A new combinatorial property for the character table of a finite group?
Let $G$ be a finite group and $\Lambda = (\lambda_{i,j})$ its character table with $\lambda_{i,1}$ the degree of the ith character.
Consider the following combinatorial property of $\Lambda$: for ...
- 44.4k
6
votes
1
answer
217
views
Number of irreducible representations of $SO_3(\mathfrak{o}/\mathfrak{p}^l)$
$\DeclareMathOperator\SO{SO}$Let $F$ be a finite extension of $\mathbb{Q}_p$, and let $\mathfrak{o}$ denote the ring of integers, with maximal ideal $\mathfrak{p}$. Let $G_l$ denote the finite group $\...
- 3,813
6
votes
1
answer
250
views
Concrete example to illustrate the theory about blocks of groups with cyclic defect groups
I'd like to to have a concrete example to illustrate the theory about blocks of groups with cyclic defect groups.
Thus, I am looking for a finite group $G$ and a prime $p$ dividing $|G|$ satisfying ...
- 44.4k
2
votes
0
answers
87
views
A theory of (or reference for) symmetric point arrangements
I wonder where I can find something written on symmetric point arrangements (see definition below). I am interested in general references, preferably books that introduce (or papers that use) some ...
- 13.6k
11
votes
1
answer
688
views
Unitary representations of finite groups over finite fields
I would like to learn the basic theory of unitary representations of finite groups over finite fields.
Here, the unitary group $\operatorname{GU}(n,\mathbb{F}_{q^2})$ consists of all invertible ...
- 37.4k
3
votes
0
answers
190
views
Efficient implementation of the Clifford group for $n$ qubits
I'm looking for an efficient implementation of the Clifford group $\mathcal{C}_n$
of $n$ qubits.
The Clifford group $\mathcal{C}_n$ has stucture $(2_+^{1+2n} \circ C_8).Sp(2,n)$,
where $2_+^{1+2n}$ ...
- 558
10
votes
1
answer
257
views
Low dimensional representations of $SL_n(\mathbb{Z}/p^\ell \mathbb{Z})$
When $\ell = 1$ I know that the smallest non-trivial irreducible complex representations of $SL_n(\mathbb{Z}/p\mathbb{Z})$ has dimension $\frac{p^n - 1}{p-1} - 1$ (with maybe some exceptions for ...
- 2,242
2
votes
2
answers
267
views
Irreducible representations of $G_4 = \langle a,b \mid a^{16}, b^{2}, baba^{-7}\rangle$ and other Semidihedral groups
I would like to know the irreducible representations of the group $G_4 = \langle a,b \mid a^{16}, b^2, baba^{-7}\rangle$ and its character table.
More than that, I would like to know the irreducible ...
- 44.4k
4
votes
0
answers
218
views
Conjugacy class representatives for the automorphism group of a finite abelian group
Given a finite abelian group $A$, I'd like a list of conjugacy class representatives for its automorphism group ${\rm Aut}(A)$.
In fact, it's not important that I have exactly one representative from ...
- 6,919
5
votes
1
answer
305
views
Schur covers of affine 2-transitive groups
I am interested in Schur covers of minimal 2-transitive groups. A theorem of Burnside gives that every finite 2-transitive group is either almost simple or affine. In the time since, these groups have ...
- 37.4k
6
votes
2
answers
856
views
Algorithm for Brauer lifting via Brauer tree?
Background: Given a finite group $G$ and a prime $p$ dividing its order, Brauer theory compares the ordinary characters of $G$ with the Brauer characters arising from $p$-modular representations. On ...
- 61
8
votes
2
answers
618
views
Reference request: Models of cuspidal representations of GL(n,k) where k is a finite field
Let $k=\mathbb{F}_q$ where $q$ is a prime power of odd cardinality.
Where could I find explicit models of all irreducible cuspidal (complex) representations of $GL_n(k)$ for $n\ge 3$?
I understand ...
- 41
6
votes
1
answer
366
views
Group of order $5p^aq^b$
In Lectures by Dan Bump on Modular representation theory,
Theorem 13.14 states that whenever $G$ is a non-abelian simple group of order $|G|=p^aq^br$ for distinct primes $p$,$q$, and $r$, every $r$-...
- 1,666
2
votes
0
answers
72
views
"Spectral gaps" of commutativity measure
There's a notion of commutativity measure $P(G)$ of a finite group $G$ which is probably folklore: count commuting pairs in $G \times G$ and divide by $|G \times G|$. There are some results:
$P(...
- 4,600
17
votes
5
answers
3k
views
Reference for this theorem in representation theory?
Let $G$ be a finite group and $\chi$ be an irreducible character of
$G$ (characteristic zero algebraically closed base field). If $H$ is
the kernel of $\chi$ then the irreducible representations of $G/...
- 44.4k
7
votes
3
answers
1k
views
the character tables of irreducible representations of $SL(3,Z_q)$
The following paper gives a classification of the character tables of irreducible representations of $SL(3,GF(q))$ where $q$ is a power of a prime number, and $ GF(q)$ a finite field of $q$ elements.
...
- 3,813
1
vote
0
answers
136
views
Representations of finite groups over commutative rings-question and reference request
In a textbook of representation theory I have encountered the following statement without proof:
Let $R$ be a commutative ring and $G$ a finite group. If $M$ is a simple $RG$-module then the ...
- 63.9k
1
vote
0
answers
98
views
Ireducible representations in characteristic p [duplicate]
It is known that the intersection of kernels of all ireducible representations of a finite group in characteristic zero is the trivial group.
In characteristic $p>0$ I have understood that this ...
- 43.2k
9
votes
5
answers
2k
views
A catalog of faithful representations of finite groups?
I want a reference that catalogs the smallest-dimensional faithful representation of every noteworthy finite group. Specifically, I want representations on $\mathbb{R}^n$ and $\mathbb{C}^n$.
Where ...
- 1,661
6
votes
2
answers
2k
views
Alternative or reprint of Carter's "Finite Groups of Lie Type: Conjugacy Classes and Complex Characters"
I would like to learn about character theory of finite groups of Lie type and some Deligne-Lusztig theory. The classic textbook on the subject seems to be Roger W. Carter's Finite Groups of Lie Type: ...
CommunityBot
- 1
8
votes
2
answers
576
views
Two statistics on the permutation group
Let $\mathfrak{S}_n$ be the permutation group on an $n$-element set. For each fixed $k\in\mathbb{N}$, consider the two sets
$$A_n(k)=\{\sigma\in\mathfrak{S}_n\vert\,\, \text{$\exists i,\,\, 1\leq i\...
- 5,822
9
votes
2
answers
485
views
Reference for restriction of a simple module over a splitting field to a smaller field?
This is mainly a request for a straightforward reference (preferably at textbook level). The question comes up while responding to a question raised by non-specialists in finite group representations....
- 1,991
7
votes
2
answers
780
views
Finite groups with a character having very few nonzero values?
A number theorist I know (who studies Galois representations) raised a question recently:
Which finite groups can have an irreducible character of degree at least 2 having only $n=2, 3$, or 4 ...
- 115
6
votes
1
answer
476
views
Structure of Deligne-Lusztig representations $R_{T,\theta}$ for ministropic $T$ and cuspidal representations
Let $G$ be a reductive group over a finite field $k$, let $F$ be a Frobenius morphism on $G$.
I'll start with a somewhat vague question and make my question more specific further down:
How do ...
- 52.9k
2
votes
0
answers
187
views
Classification of Automorphism set of a Regular graph
Let $A$ be the adjacency matrix of an $r$-regular graph $G$ with $n$ vertices (Not complete or cycle graph) . Also, let $Aut(G)$ be the set of all its automorphisms (i.e. set of permutation matrices)....
CommunityBot
- 1
3
votes
0
answers
102
views
Localized at $p$ integral representations of finite elementary $p$-groups
Let $C_p$ be a cyclic group of prime order $p$.
Let $F=C_p^n=C_p\times\dots\times C_p$ ($n$ times).
I would like to to classify finite dimensional representations of $F$ over ${\mathbb{Z}}$.
However, ...
- 14.2k
2
votes
2
answers
535
views
Is anything known about the eigenspectrum of the regular representation of the permutation group?
I am looking for information like upper bounds on how many times any eigenvalue can occur or something like how many eigenvalues can be there in some given range. Is anything like this known?
The ...
CommunityBot
- 1
3
votes
0
answers
282
views
Galois correspondence subgroups/subsystems
In this paper (1998) by M. Izumi, R. Longo, S. Popa, there is the following result (page 49) on compact groups:
Lemma 3.16. Let $G$ be a compact group and $Rep(G)$ the category of finite ...
3
votes
1
answer
814
views
Reference for Clifford theory of algebras
Clifford theory relates the representation theory of a group to that of a normal subgroup. A good reference for this is Curtis and Reiner's "Methods in Representation theory II", Theorem 11.1.
...
- 6,210
9
votes
2
answers
933
views
Good effective versions of theorems of Artin and Brauer
The theorem of Artin and Brauer of the title are the famous theorem in the theory of representation of finite groups.
For example, Artin's theorem is the statement that for every character $\chi$ of ...
- 44.4k
2
votes
0
answers
89
views
explicit matrices for Weil ($p^2$ dimensional) representation of $Sp(4,\mathbb{F}_p)$, $p>3$
I am looking for more-or-less explicit matrices for the $p^2$ dimensional Weil representation of $Sp(4,\mathbb{F}_p)$, suitable for computer implementation. Ideally, I would like the images of the ...
- 1,639