All Questions
Tagged with rt.representation-theory gr.group-theory
149 questions
42
votes
7
answers
10k
views
Bijection between irreducible representations and conjugacy classes of finite groups
Is there some natural bijection between irreducible representations and conjugacy classes of finite groups (as in case of $S_n$)?
- 1,318
13
votes
2
answers
1k
views
Number of commuting pairs (triples, n-tuples) in GL_n(F_q) (and other groups)?
Question 1 What is the number of pairs of commuting elements in GL_n(F_q) ?
I am aware of many results concerning commuting elements in Mat_n(F_q), but I am interested in GL i.e. non-degenerate ...
- 24.9k
35
votes
2
answers
3k
views
Examples of finite groups with "good" bijection(s) between conjugacy classes and irreducible representations?
For symmetric group conjugacy classes and irreducible representation both are parametrized by Young diagramms, so there is a kind of "good" bijection between the two sets. For general finite groups ...
- 24.9k
20
votes
1
answer
586
views
$q$-(and other)-analogs for counting index-$n$ subgroups in terms of Homs to $S_n$?
The following formula of astonishing beauty and power (imho):
$$ \sum_{n \ge 0} \frac{| \mathrm{Hom}(G,S_n) | }{n! } z^n = \exp\left( \sum_{n \ge 1} \frac{|\text{Index}~n~\text{subgroups of}~ G|}nz^...
- 24.9k
12
votes
3
answers
1k
views
Is there a purely group-theoretic reformulation of an equivalence of subgroups?
There is an equivalence relation between inclusion of finite groups coming from the world of subfactors:
Definition: $(H_{1} \subset G_{1}) \sim(H_{2} \subset G_{2})$ if $(R^{G_{1}} \subset R^{H_{1}}...
23
votes
2
answers
2k
views
Orbit structures of conjugacy class set and irreducible representation set under automorphism group
let G be a finite group. Suppose C is the set of conjugacy classes of G and R is the set of (equivalence classes of) irreducible representations of G over the complex numbers.
The automorphism group ...
- 7,320
13
votes
1
answer
2k
views
A dual version of a theorem of Øystein Ore in group theory
This post is a dual version for the Generalization of a theorem of Øystein Ore in which it's proved:
Theorem: Let $[H, G]$ be a distributive interval of finite groups. Then $\exists g \in G$ such ...
78
votes
3
answers
10k
views
5/8 bound in group theory
The odds of two random elements of a group commuting is the number of conjugacy classes of the group
$$ \frac{ \{ (g,h): ghg^{-1}h^{-1} = 1 \} }{ |G|^2} = \frac{c(G)}{|G|}$$
If this number exceeds ...
- 22.8k
43
votes
0
answers
2k
views
Why are there so few quaternionic representations of simple groups?
Having spent many hours looking through the Atlas of Finite Simple Groups while in Grad school, I recall being rather intrigued by the fact that among the sporadic groups, only one (McLaughlin as I ...
- 5,191
33
votes
2
answers
1k
views
Analogies supporting heuristic: Weyl groups = algebraic groups over field with one element?
There is well-known heuristic that Weyl groups are reductive algebraic groups over "field with one element".
Probably the best known analogy supporting that heuristic is the limit $q\to1$ ...
- 24.9k
4
votes
3
answers
2k
views
Representation theory of p-groups in particular upper tringular matrices over F_p
Finite p-groups - have p^n elements by definition. According to WP there is rich structure theory.
Question: How far is representation theory of p-groups is understood?
In case this question is too ...
- 24.9k
52
votes
2
answers
5k
views
Is there a good mathematical explanation for why orbital lengths in the periodic table are perfect squares doubled?
$\DeclareMathOperator\SO{SO}\newcommand{\R}{\mathbb{R}}\newcommand{\S}{\mathbb{S}}$The periodic table of elements has row lengths $2, 8, 8, 18, 18, 32, \ldots $, i.e., perfect squares doubled. The ...
- 541
51
votes
2
answers
4k
views
Which philosophy for reductive groups?
I am just beginning to look further into trace formulas and automorphic forms in a quite general setting. For long I have noticed that the natural assumption on the group $G$ we work on is to be ...
- 5,424
32
votes
3
answers
3k
views
Order of products of elements in symmetric groups
Let $n \in \mathbb{N}$. Is it true that for any $a, b, c \in \mathbb{N}$ satisfying
$1 < a, b, c \leq n-2$ the symmetric group ${\rm S}_n$ has elements of order $a$ and $b$
whose product has order $...
- 19.6k
31
votes
1
answer
2k
views
Can one explain Tannaka-Krein duality for a finite-group to ... a computer ? (How to make input for reconstruction to be finite datum?)
Consider a finite group. Tannaka-Krein duality allows to reconstruct the group from the
category of its representations and additional structures on it (tensor structure + fiber functor). Somehow ...
- 24.9k
23
votes
7
answers
7k
views
Faithful representations and tensor powers
The following result was mentionned earlier in this thread, I searched a bit in the related threads and couldn't find a proof. I would really like to see a proof of it:
Let $G$ be a finite group and $...
- 231
17
votes
0
answers
692
views
Monstrous Langlands-McKay or what is bijection between conjugacy classes and irreducible representation for sporadic simple groups?
Context: The number of conjugacy classes equals to the number of irreducuble representations (over C) for any finite group. Moreover for the symmetric group and some other groups there is "good ...
- 24.9k
9
votes
2
answers
526
views
Strongly real elements of odd order in sporadic finite simple groups
Recall that an element of a finite group is said to be real if it is conjugate to its inverse, and strongly real if the conjugating element can be chosen to be an involution.
Question: Is it true ...
- 1,090
8
votes
4
answers
2k
views
Product of conjugacy classes - is there an analog of Tanaka-Krein reconstruction ?
Consider a finite group G. The product of conjugacy classes can be defined in natural way just by multiplying the representatives and counting multiplicities (see e.g. MO 62088). So we get ring with ...
- 24.9k
6
votes
4
answers
3k
views
Dixmier's lemma as a generalisation of Schur's first lemma
I thought that this question is simple, and asked it at Stackexchange. To my surprise, no one was able to answer it there. Now have to elevate it to Overflow.
What mathematicians call Schur's lemma ...
- 603
5
votes
5
answers
1k
views
Irreducible representations of the unitriangular group
Hi,
I wonder how much is known about the irreducible representations of the nxn unitriangular group over a finite field with q elements.
I know that all characterdegrees are a power of q and all ...
- 891
45
votes
1
answer
5k
views
Square roots of elements in a finite group and representation theory
Let $G$ be a finite group. In an an earlier question, Fedor asked whether the square root counting function $r_2:G\rightarrow \mathbb{N}$, which assigns to $g\in G$ the number of elements of $G$ that ...
- 13k
44
votes
10
answers
11k
views
The finite subgroups of SL(2,C)
Books can be written about the finite subgroups of $\mathrm{SL}(2,\mathbb C)$ (and their immediate family, like the polyhedral groups...) I am about to start writing notes for a short course about ...
- 47.3k
43
votes
3
answers
10k
views
Feit-Thompson theorem: the Odd order paper
For reference, the Feit-Thompson Theorem states that every finite group of odd order is necessarily solvable. Equivalently, the theorem states that there exist no non-abelian finite simple groups of ...
- 1,993
28
votes
4
answers
3k
views
Groups in which all characters are rational.
The Symmetric groups $S_n$ has interesting property that all complex irreducible characters are rational (i.e. $\chi(g)\in \mathbb{Q}$ for all $\mathbb{C}$-irreducible characters $\chi$,$\forall g\in ...
- 427
26
votes
1
answer
816
views
What are the points of simple algebraic groups over extensions of $\mathbb{F}_1$?
The "field with one element" $\mathbb{F}_1$ is, of course, a very speculative object. Nevertheless, some things about it seem to be generally agreed, even if the theory underpinning them is not; in ...
- 32.5k
20
votes
5
answers
2k
views
How small can a group with an n-dimensional irreducible complex representation be?
More precisely, what is the smallest exponent e such that, for every n, there exists a group of size at most Cn^e for some absolute constant C and with an n-dimensional irreducible complex ...
- 118k
20
votes
0
answers
451
views
Row of the character table of symmetric group with most negative entries
The row of the character table of $S_n$ corresponding to the trivial representation has all entries positive, and by orthogonality clearly it is the only one like this.
Is it true that for $n\gg 0$, ...
- 24.2k
20
votes
2
answers
948
views
The finite groups with a zero entry in each column of its character table (except the first one)
$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\Aut{Aut}$Consider the class of finite groups $G$ having a zero entry in each column of its character table (except the first one), i.e. for all $g \...
19
votes
3
answers
2k
views
A character identity
This is related to my question, but it concerns a specific point of the proof of Schur's Theorem.
Let $G$ be a finite group and $\chi$ an irreducible character of $G$. Is it true that
$$\forall g\in ...
- 52.3k
17
votes
1
answer
1k
views
Why do these two Monster-related calculations yield $163$?
Fact 1: (1979, Conway and Norton)$^{1}$
"There are $194-22-9=\color{blue}{163\,}$ $\mathbb{Z}$-independent McKay-Thompson series for the Monster."
Note: There are 194 (linear) irreducible ...
- 12.6k
11
votes
1
answer
569
views
Counting symmetric subgroups of symmetric groups
This question is related to, but much more specific than, this one.
For $k \leq n$, let $a(k,n)$ denote the number of conjugacy classes of subgroups of the symmetric group $S_n$ which are isomorphic ...
- 2,753
11
votes
5
answers
2k
views
Structure of the adjoint representation of a (finite) group (Hopf algebra) ?
Every group acts on itself by conjugation $h \mapsto g h g^{-1}$. Respectively considering functions on a group we obtain a linear representation.
Question 1: what is known about this representation ...
- 24.9k
9
votes
1
answer
683
views
Is $\operatorname{PSL}(2,q)$ the most quasirandom group?
Is the following statement true?
Every finite group $G$ has a non-trivial irreducible representation of dimension $O(\lvert G\rvert^{1/3})$.
Context: Groups with no small irreducible representations ...
- 986
9
votes
1
answer
277
views
Algorithmically handling the Spin groups in larg(ish) dimensions
Question: Is there a reasonably efficient algorithmic representation of $\mathit{Spin}_n$? By this I mean, a way to store its elements and operate on them (multiply, inverse, maybe compute ...
- 32.5k
6
votes
1
answer
2k
views
Explicit description of all morphisms between symmetric groups.
There is a well-known morphism $S_4\to S_3$, obtained by having $S_4$ act on the three partitions of $4$ objects into $2+2$. Similarly, given any $n$, one can devise a morphism $S_n\to S_k$ for some $...
- 14.4k
6
votes
1
answer
339
views
A property forcing the Frobenius-Schur indicator to be positive
Let $G$ be a finite group. Two irreducible complex representations $V,V'$ of $G$ are called dual to each other if $V \otimes V'$ admits a trivial component, i.e. $\hom_G(V \otimes V',V_0)$ is positive ...
5
votes
2
answers
1k
views
Decomposing representations of finite groups
Let $G$ be a finite group, $p$ a prime number. We denote by $\mathbb{F}_p$ the field of cardinality $p$. Let $V$ be an infinite dimensional representation of $G$ over $\mathbb{F}_p$.
Must there be $G$...
- 11.3k
5
votes
2
answers
663
views
Decomposing the conjugacy representation of Sym$(n)$ for small $n$
I am trying to compute the decomposition of the conjugacy representation of some small symmetric groups. Perhaps someone has undertaken a similar calculation.
My own calculations are quite slow, ...
- 1,081
3
votes
3
answers
461
views
A problem with pointwise stabilizer subgroups of fixed-point subspaces II
Definitions: Let $W$ be a representation of a group $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$.
Let the fixed-point subspace $W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$.
Let ...
3
votes
1
answer
506
views
Is there a link between $H_2(G,\mathbb{Z})$, the Schur Multiplier of a group, and the "other" Schur multipliers of a group?
The name for the the following 2 mathematical objects:
$$H_2(G,\mathbb{Z})$$ and
$$\{K:G\times G\longrightarrow\mathbb{C}\ |\ \forall T\in B(l^2(G))\text{we have that}~S:G\times G\longrightarrow\...
- 647
3
votes
1
answer
257
views
Irreducibles of virtually abelian finitely generated groups
I have a few related questions. First I would appreciate it if someone could provide me with a reference for the following
"Complex unitary irreducibles of virtually abelian groups have bounded ...
- 193
2
votes
1
answer
298
views
An upper bound for the maximal subgroups at fixed index?
Let us call a subgroup an injective homomorphism between groups.
I warn the reader that a subgroup designates here an inclusion $(H \subset G)$, not $H$ alone.
A subgroup $H \subset G$ is ...
0
votes
1
answer
473
views
Projective characters with corresponding factor set
The following is just a follow up to my previous question. I have a finite group $H$ with 14 ordinary characters. The Schur multiplier $M(H)\cong 2^2$. Hence the group $H$ will have 3 sets of ...
63
votes
1
answer
4k
views
Feit-Thompson conjecture
The Feit-Thompson conjecture states:
If $p<q$ are primes, then $\frac{q^p-1}{q-1}$ does not divide $\frac{p^q-1}{p-1}$.
On page xiii of these proceedings of a conference at the University of ...
- 26.5k
55
votes
5
answers
6k
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 ...
- 35.5k
38
votes
0
answers
1k
views
Groups whose complex irreducible representations are finite dimensional
By a complex irreducible representation of a group $G$, I mean a simple $\mathbb CG$-module. So my representations need not be unitary and we are working in the purely algebraic setting.
It is easy ...
- 38.6k
35
votes
4
answers
2k
views
Being a subgroup: proof by character theory
Let me first cite a theorem due to Frobenius:
Let $G$ be a finite group, with $H$ a proper subgroup ($H\ne (1)$ and $G$). Suppose that for every $g\not\in H$, we have $H\cap gHg^{-1}=(1)$. Then
$...
- 52.3k
27
votes
3
answers
2k
views
How can classifying irreducible representations be a "wild" problem?
Let $q$ be a prime power and $U_n(\mathbb{F}_q)$ be the group of unitriangular $n\times n$-matrices. I've read and heard in several places (see e.g. this mathoverflow question) that classifying ...
- 2,450
25
votes
7
answers
8k
views
Applications of group theory to mathematical biology (pharmacology)
Are there applications of group theory — broadly, say, representation theory, Lie algebras, $q$-groups, etc — to mathematical biology?
In particular, I am interested in applications to pharmacology — ...