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19 votes
1 answer
4k views

How should I think about the module of coinvariants of a $G$-module?

Let $G$ be a group, $M$ a $G$-module, then the group of coinvariants is the module $M_G := M/I_GM$, where $I_G$ is the kernel of the augmentation map $\epsilon : \mathbb{Z}G\rightarrow \mathbb{Z}$. ...
stupid_question_bot's user avatar
15 votes
4 answers
869 views

What is known about ordinary character values at involutions?

Let $G$ be a finite group and let $\chi$ be the character of an irreducible complex representation $\rho$ of $G$ on $V$. Let $x$ be an involution in $G$. I'd like to ask the following Question 1: ...
Bernhard Boehmler's user avatar
12 votes
2 answers
984 views

Common basis for permutation matrices

How can I check whether there exists a common basis with respect to which two matrices 𝐴 and 𝐵 are permutation matrices? More explicitly, let $A$ and $B$ be two unitary matrices whose eigenvalues ...
as2457's user avatar
  • 295
9 votes
3 answers
350 views

$G$-module structure of the relation module for a presentation of a finite group $G$

Let $F_n$ be a free group of rank $n\ge 2$, and $F_n\rightarrow G$ a surjection with $G$ finite. Let $R$ be the kernel. From this, we get an action of $G$ on the abelianization $R/R'$ (a free abelian ...
stupid_question_bot's user avatar
9 votes
2 answers
814 views

Regular elementary abelian subgroups of primitive permutation groups

A finite group $B$ is said to be a B-group if every primitive permutation group having a regular (transitive) subgroup isomorphic to B is $2$-transitive. Schur proved that a cyclic group of ...
Mark Wildon's user avatar
  • 11.2k
8 votes
2 answers
2k views

Expectation of trace of nth power of unitary matrices

I am trying to find the answer of $$\int dU \ |Tr(U^m)|^2$$ where $m\in\mathbb{N}$ and $U$'s are unitary matrices in $\textit{U}(n)$ and $dU$ is a normalized Haar measure. In the case $m=1$, the ...
Atnap's user avatar
  • 127
7 votes
1 answer
326 views

Independent vectors in the permuting coordinates action of $S_n$ on $\mathbb{R}^n$

Let $V$ be the hyperplane in $\mathbb{R}^n$ with equation $\sum_i x_i=0$. The symmetric group $S_n$ acts on $V$ by $s\cdot (v_1,\ldots,v_n)=(v_{s^{-1}(1)},\ldots,v_{s^{-1}(n)})$. Consider those $v\in ...
Brent Everitt's user avatar
6 votes
1 answer
205 views

Can a projective solvable group be transitive?

Let $p > 3$ be a prime number, and let $G \leq \mathrm{PGL}_2(\mathbb{F}_p)$ be a solvable subgroup. Is it possible that the action of $G$ on $\mathbb{P}^1(\mathbb{F}_p)$ is transitive?
Pablo's user avatar
  • 11.3k
6 votes
1 answer
934 views

Finite groups with all irreducible representations one dimensional

Let $k$ be a field of characteristic $p$ and $G$ a finite group. This question might be a dulicate of this question: Which finite groups have no irreducible representations other than characters? ...
user103474's user avatar
5 votes
2 answers
387 views

Presentations of $\mathbf{PGL}_3(\mathbb{F}_2)$ by three involutions

I am searching for (two) presentations of the group $\mathbf{PGL}_3(\mathbb{F}_2)$ for which the generators are involutions $a, b, c$, and such that the following relations are present [among extra ...
THC's user avatar
  • 4,547
5 votes
1 answer
507 views

Matrix Elements of Real Representations

I asked this question over at Math.StackExchange and despite having had a bounty on it I did not receive an answer. Suppose that $G$ is a finite group and we have a unitary irreducible representation ...
JP McCarthy's user avatar
  • 1,037
5 votes
1 answer
908 views

Finding a basis for the (linear combinations) span of a matrix group, efficiently?

I have an algorithm whose bottleneck is the following task: Let $\mathbb{F}$ be a finite field. Given a set of $k$ invertible matrices $g_1,\dots,g_k\in GL_n(\mathbb{F})$, let $G=\langle g_1,\dots,...
Boaz Tsaban's user avatar
  • 3,104
4 votes
1 answer
365 views

Classes of finite resoluble groups which are (faithfully) representable by triangular matrices?

Let $G$ be a group, $k$ a field and $T(n,k)\subset Gl(n,k)$, the group of invertible upper triangular $n\times n$ matrices. I know that if $\rho : G\rightarrow T(n,k)$ is faithful (i.e. into) then $...
Duchamp Gérard H. E.'s user avatar
4 votes
1 answer
298 views

Characterizations of groups whose general linear representations are all trivial

Let $G$ be a group. Suppose for any general linear representation $\rho:G\to\mathrm{GL}(n)$, $\rho$ must be trivial. Question: Are there any characterizations or equivalent conditions for $G$? Thanks ...
Shiquan Ren's user avatar
  • 1,990
4 votes
1 answer
297 views

Which groups can be reconstructed from a single invariant subspace?

Let $G\subseteq\mathrm{Perm}(\Bbb R^n)$ be a matrix group consisting of permutation matrices acting on $\Bbb R^n$. Let $U\subseteq\Bbb R^n$ be an irreducible invariant subspace w.r.t. $G$. Now, define ...
M. Winter's user avatar
  • 13.6k
4 votes
2 answers
433 views

What is the current status of representation theory of $n$-ary groups in terms of hypermatrices?

An $n$-ary group is a generalization of the usual concept of a group where the binary operation (2-argument operation) is instead an $n$-ary ($n$-argument) operation. More info here on Wikipedia. I ...
Sidharth Ghoshal's user avatar
4 votes
1 answer
513 views

Checking irreducibility

This is related to this question. Suppose I have an $n$-dimensional representation of a finitely generated group, and I want to know whether it is absolutely irreducible. This can, of course, be done ...
Igor Rivin's user avatar
  • 96.4k
4 votes
0 answers
219 views

Map $\operatorname{Sym}^{mp}(V^*) \longrightarrow K^{q}$ defined by $q$ points in $\operatorname{Sym}^p(V)$

EDIT : I have edited the question and made it more specific with respect to the kind of answer I expect. Let $V$ be a finite dimensional $K$-vector space and let $x_1, \dotsc, x_q \in V$ be $q$ points,...
Libli's user avatar
  • 7,300
4 votes
0 answers
164 views

Matrices in $SL(2,\mathbb{C})$ with characteristic polynomial defined over a subring

Let $R\subset\mathbb{C}$ be a subring, and let $A,B\in SL(2,\mathbb{C})$ be matrices such that $A,B,AB$ all have trace in $R$. For which $R$ can we then deduce that $A,B$ are simultaneously conjugate ...
stupid_question_bot's user avatar
3 votes
3 answers
1k views

A table for irreducible integral representation of finite cyclic groups

Is there such a table where the irreducible integral representations of finite cyclic groups are listed? Edited: Thanks for Todd Leason's comment.Acutally,i want to know all inequivalent ...
user2015's user avatar
  • 593
3 votes
3 answers
241 views

For centralizer subgroups, is the endomorphism ring of a restriction generated by endomorphisms and the centralized element?

In some recent doodlings, I got myself to the point where what I was trying to understand would work out if the following claim were true: Let $G$ be a group, $g\in G$, and $\rho:G \to \...
Theo Johnson-Freyd's user avatar
3 votes
1 answer
527 views

An expectation of the product of random unitaries

I want to find the answer of $$\int dU \ U^m X \ U^{\dagger m}$$ Where $m\in\mathbb{N}$ and $U$'s are unitary matrices in $U(n)$ and $dU$ is a normalized Haar measure. $X$ is a given self-adjoint ...
Atnap's user avatar
  • 127
3 votes
3 answers
190 views

Simultaneous "Monomialization" of a set of operators.

We all know that a set of commuting diagonalizable matrices can be simultaneously put in diagonal form. My general question is: Under what conditions can a set of (diagonalizable) matrices be ...
Eric Rowell's user avatar
  • 1,639
3 votes
1 answer
237 views

invariant subspaces of general linear groups for finite fields

Let $K$ be a finite field, let $n\ge 1$ be an integer and let $G=\mathrm{GL}(n,K)$ be acting linearly on a finite dimensional $K$-vector space $V$. Although $G$ is a reductive group, it is not ...
Jérémy Blanc's user avatar
3 votes
2 answers
412 views

Indecomposable integral representations of a group of order 2 "by hand"

This question is a duplicate of that 2010 MO question. I am interested in classifying isomorphism classes of $n$-dimensional integral representations of the cyclic group $C_2$ of order $2$. Clearly, ...
Mikhail Borovoi's user avatar
2 votes
1 answer
419 views

Heisenberg group over the Gaussian integers

If we take the entries of the (standard $3 \times 3$) Heisenberg group to live in the Gaussian integers $\mathbb{Z}[i]$, what is the structure of this group? Are all of its representations known?
Dr Shello's user avatar
  • 1,180
2 votes
0 answers
101 views

On the irreducible submodules of adjoint representations $\text{ad}^{0}$

Let $k$ be a finite field of characteristic $p$. Let $H$ be a subgroup of $\rm{GL}_{n}(k)$ of order prime to $p$ where $n\geq2$. Assume that the representation $H\hookrightarrow \rm{GL}_{n}(k)$ is ...
stupid boy's user avatar
2 votes
0 answers
75 views

Constructing representations of a topological group from characteristic polynomials of a generating set

Given a topological group $G$ and a subset $S$ of $G$ that topologically generates it, what are the conditions under which an $n$-dimensional continuous linear representation of $G$ over an ...
SHS's user avatar
  • 21
2 votes
0 answers
51 views

Relations among hyperplane mirror symmetries

Let $H(n) \subset O(n)$ be the set of mirror symmetries in $\mathbb{R}^n$ with respect to $(n - 1)$-planes containing the origin. One can see that for any $a, b, c \in H(2)$ we have $abcabc = id$, ...
Mikhail Tikhomirov's user avatar
2 votes
0 answers
63 views

Determining subgroup of finite group of Lie type from characteristic polynomials

Suppose you have $G$ a finite group of Lie type (say $\mathrm{Sp}_4( \mathbb{F}_5)$ as a case I particularly care about, but there are others.) Your friend picks a subgroup $H$ and selects random ...
Watson Ladd's user avatar
  • 2,429
1 vote
1 answer
308 views

Suppose that $G$ is a subgroup of $GL_n(\mathbb C)$ with finite exponent. Then is $G$ a finite group? [closed]

As title. the exponent of $G$ is the least number $n$ (if exists) such that $g^n=e$ holds for all $g\in G$ or $+\infty$.
Censi LI's user avatar
  • 403
1 vote
1 answer
498 views

Projectors onto the invariant subspaces of a unitary representation $U \otimes U^* \otimes U \otimes U^*$

Let $$U \mapsto U \otimes U^* \otimes U \otimes U^*$$ be a unitary representation of the unitary group $U(n)$ acting on the vector space $V$ (where $U^*$ is the complex conjugate of $U$). We can ...
Atnap's user avatar
  • 127
1 vote
0 answers
206 views

About the question "Tannaka–Krein duality"

I saw this post recently: Tannaka–Krein duality I have this question please: in the following which I report here: The problem is with surjectivity: let us denote $\mathcal{G}:=\mathcal{G}(\mathcal{R}...
user502786's user avatar
0 votes
1 answer
274 views

when $g^*$ is invariant under $Ad(G)$?

Let $G$ be a Lie Group and $\mathfrak{g}$ be its lie algebra. Let $\mathfrak{g}$ is semisimple or reductive lie algebra, then prove that $\mathfrak{g}^*$ (dual of $\mathfrak{g}$)is invariant under $...
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