All Questions
112 questions
1
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172
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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 ...
- 591
5
votes
0
answers
140
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Classification of visible actions for *reducible* representations?
Is there a classification of the pairs $(G,V)$ such that $G$ is reductive [and connected, if you like], and $G$ acts faithfully and visibly on $V$ - crucially, including all cases where $V$ is ...
- 3,230
2
votes
0
answers
118
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What are the finite-dimensional irreducible unitary representations of $E(3)$?
Let $E(3)$ be the Euclidean group of $\mathbb{R}^3$ defined, e.g., by
$$E(3)=SO(3)\ltimes T(3)$$
where $T(3)$ is the translation group.
I am looking for a reference classifying all the finite-...
- 31
16
votes
3
answers
1k
views
Conjectures in the representation theory of the symmetric group
Question: What are current open conjectures about the representation theory of the symmetric group?
I am interested mostly in the characteristic 0 case, but conjectures for the modular case can also ...
- 26.5k
1
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0
answers
110
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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
1
vote
1
answer
232
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Transfer for the group of coinvariants: a reference request
Let $G$ be a group and $M$ be a $G$-module,
that is, an abelian group written additively on which $G$ acts:
$$ (g,m)\mapsto g m.$$
We consider the group of coinvariants
$$ M_G:=G/\langle g m -m\ |\ g\...
- 14.2k
9
votes
0
answers
254
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An identity for characters of the symmetric group
I am looking for a reference for the identity
$$\chi_\lambda(C)=\frac{\dim(V_\lambda)}{|C|}\sum_{p\in P_\lambda,\,q\in Q_\lambda,\,pq\in C}\operatorname{sgn}(q)$$
for the irreducible characters of the ...
- 2,306
2
votes
0
answers
137
views
$p$-adic Banach group algebra
Let $G$ be a discrete group. Consider the Banach $\mathbb{Z}_p$-algebra: $$c_0(G, \mathbb{Z}_p) = \{ F : G \to \mathbb{Z}_p \mid \lim_{g \to \infty} |F(g)|_p = 0 \}$$ with the product given by the ...
- 1,485
1
vote
1
answer
214
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Conditions of P for existence of orthogonal matrix Q and permutation matrix U satisfying QP = PU
Question: Let $P\in \mathbb{R}^{d\times n}$ be a $d$-rank real matrix and $PP^T = c I_d$ with a certain constant $c > 0$. Under what additional conditions of $P$ does there exist an orthogonal ...
- 187
2
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0
answers
88
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Simple modules and trivial source modules
Let $(K,\mathcal{O},k)$ be a large enough $p$-modular system.
In this context, I would like to ask what is known about the following question:
when are simple $kG$-modules trivial source modules?
So ...
- 1,755
16
votes
1
answer
408
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Is there a relationship between Broué's abelian defect group conjecture and Alperin's weight conjecture?
Let $G$ be a finite group, let $k$ be a large enough field of characteristic $p>0$. Let $p\mid |G|$.
Broué's abelian defect group conjecture states the following:
Let $B$ be a block of $kG$ with
...
- 1,755
1
vote
0
answers
83
views
$p$-modular splitting systems and the characteristic of the ring $\mathcal{O}$
Let $k=\overline{k}$ be a field of characteristic $p$.
Let $(K,\mathcal{O},k)$ be a $p$-modular system.
Let both $k$ and $K$ be splitting fields for $G$ and its subgroups.
The ring $\mathcal{O}$ can ...
- 311
6
votes
0
answers
236
views
Group homomorphism from $\mathrm{GL}_p$ to $\mathrm{SL}_p$ in characteristic $p$
If $k$ is a commutative field of characteristic $p>0$, then the map
$$ \theta \colon \mathrm{GL}_p(k) \to \mathrm{SL}_p(k) \colon A = (a_{ij}) \mapsto (\det A)^{-1} (a_{ij}^p) $$
is a group ...
- 6,614
4
votes
2
answers
378
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Splitting field for $\mathrm{GL}(2,p)$ - reference request
It seems to me from a quick glance at several sources describing the complex and modular irreducible representations of $\mathrm{GL}(2,p)$ that any field $K$ containing a primitive $(p-1)$-root of ...
- 38.6k
3
votes
0
answers
463
views
Representations of triangle groups
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\PSL{PSL}$I am self-studying triangle groups and the following question comes up.
Let $G$ denotes $(2,3,7)$ triangle group. It is symmetry group of $(2,...
- 613
2
votes
0
answers
98
views
Question concerning relationships between different $p$-modular systems and Brauer character values
Let $(K,\mathcal{O},k)$ be a large enough $p$-modular system, where $\mathcal{O}$ is a complete discrete valuation ring of characteristic zero with unique maximal ideal $J(\mathcal{O})$, algebraically ...
- 1,755
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 ...
- 249
18
votes
2
answers
1k
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Has anyone seen this construction of the Weil representation of $\mathrm{Sp}_{2k}(\mathbb{F}_p)$?
$\def\FF{\mathbb{F}}\def\CC{\mathbb{C}}\def\QQ{\mathbb{Q}}\def\Sp{\text{Sp}}\def\SL{\text{SL}}\def\GL{\text{GL}}\def\PGL{\text{PGL}}$Let $p$ be an odd prime. The Weil representation is a $p^k$-...
- 156k
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$, $...
1
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0
answers
287
views
Characters of upper triangular matrices over finite field - reference request
Let $B_n$ be the group of upper matrices and $U_n$ the subgroup of unipotent upper triangular matrices. I would like some references which discusses complex character theory of $B_n(\mathbb{F}_q)$ for ...
- 2,751
9
votes
1
answer
435
views
Questions on the group $\mathrm{GL}(H)$
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\U{U}$Let $H$ be an infinite dimensional complex Hilbert space. Consider the group $\GL(H)$ of bounded invertible operators on $H$.
Question 1. I've ...
- 1,586
7
votes
1
answer
281
views
Question concerning the coefficients of block idempotents
Let $G$ be a finite group. Let $p$ be a prime number such that $p \mid |G|$.
Let Irr$(G)$ denote the set of ordinary irreducible characters of $G$.
For $\chi\in$ Irr$(G)$ define $e_{\chi} := \frac{\...
- 1,755
3
votes
0
answers
205
views
Status of RFD groups and $C^*$-algebras
Motivated by this question and its great answers, I become very curious to know what do we know about RFD (residually finite dimensional) groups and $C^*$-algebras, e.g. do we know how these ...
- 1,586
7
votes
1
answer
778
views
Reference for the Brauer-Nesbitt theorem
In Wikipedia's article on the Brauer-Nesbitt theorem, they state that given a group $G$ and a field $E$, two semisimple representations $\rho_1,\rho_2 : G\longrightarrow \operatorname{GL}_n(E)$ are ...
- 7,527
4
votes
1
answer
197
views
Moment integrals and determinants
Let $USp(2n)$ be the compact symplectic group of size $2n$, $dA$ its Haar measure
of total mass one, and $\det(1−A)$ being computed for the standard representation of
$A\in USp(2n)$ as a matrix of ...
- 43.2k
7
votes
2
answers
713
views
Total sum of squares of characters of the symmetric group $\mathfrak{S}_n$
In my earlier MO post, I proposed the double sum $\sum_{\mu\vdash n}\sum_{\lambda\vdash n}\chi_{\mu}^{\lambda}$ regarding characters of the symmetric group $\mathfrak{S}_n$. Soon after, I started ...
- 43.2k
3
votes
2
answers
449
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Is there any simple formula for the character of $S_{n}$ represented by the set of $k$-tuples of $\{1,2,...,n\}$?
I'm interested in the representation theory of symmetric groups.
I'm now trying to search for the formula for the characters of $\Omega^{k}$, the set of $k$-tuple of elements of $\Omega$ a set of $n$ ...
- 1,013
3
votes
0
answers
115
views
Reference for the Netto's theorem on the permutation groups which was mentioned in the paper of Frobenius
I'm trying to read 'Uber die Charaktere der mehrfach transitiven Gruppen' written by Frobenius.
There he mentioned some theorems of Netto.
I'm depending on the Google translator. and the translation ...
- 1,013
8
votes
2
answers
482
views
Parabolics and simple roots for a special unitary group: reference request
I am looking for a reference where the relative root system, the relative system of simple roots, and parabolic $\Bbb R$-subgroups for the real algebraic group ${\rm SU}(p,q)$ are explicitly computed.
...
- 14.2k
6
votes
1
answer
252
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Is there a known classification of regular multiplicity-free permutation groups?
The question is in the title, but let me clarify the terminology. I consider a permutation group $\Sigma\subseteq\mathrm{Sym}(\Omega)$ on a finite set $\Omega$.
$\Sigma$ is regular if it acts ...
- 13.6k
0
votes
0
answers
181
views
Request for a modern Reference for Frobenius' paper "Über die Charaktere der mehrfach transitiven Gruppen"
I'm interested in the paper of Jan Saxl "The Complex Characters of the Symmetric Groups that Remain Irreducible in Subgroups".
I have only (not yet enough!) standard background on the ...
- 1,013
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
6
votes
1
answer
186
views
Endomorphism ring of trivial source modules for abelian p-groups
Bernhard Böhmler (who is also on MO) and myself had the following idea:
Let $G$ be a finite group and $k$ a field of characteristic $p$ (algebraically closed when it is needed) such that $p$ divides ...
- 26.5k
1
vote
0
answers
107
views
Reference request concerning splitting fields for groups that are related to special symmetric groups
Denote the symmetric group of order $n!$ by $S_n$. Let $H:=S_p$ for an odd prime $p$.
Every finite field $k$ is a splitting field $(^*)$ for $kH$, in particular $k:=\mathbb{F}_p$.
Questions:
Is $k:=\...
- 311
8
votes
1
answer
513
views
Equivariant (co)homology of flag manifolds, convolution algebra and nil hecke algebra?
For a complex reductive group $G$ and its Borel subgroup $B$, it seems to be well-known that the equivariant homology group $H^G_*(G/B\times G/B)$ forms a nil-Heck algebra
$$NH=\Bbbk[y_i,\partial_{j}]...
- 719
2
votes
0
answers
99
views
Character degrees in induced blocks
Let $G$ be a finite group and $U\leq H\leq G$ a chain of subgroups.
Presume that $p$ is a prime dividing the order of $U$.
Suppose that $b_1$ is a $p$-block of $U$ and $b_2$ a $p$-block which is ...
- 311
3
votes
2
answers
279
views
Searching for theorems characterizing when $O_p(G)$ is trivial / non-trivial
Let $G$ be a finite group. Let $p$ be a prime.
Let $O_p(G)$ be the $p$-core of $G$.
Are there any theorems known saying something like
$O_p(G)$ is trivial, if and only if ... and
$O_p(G)$ is non-...
- 237
3
votes
1
answer
95
views
Question concerning Brauer's second main theorem, Brauer correspondent blocks and blocks covered by nilpotent blocks
A version of Brauer's second main theorem is as follows:
Let $G$ be a finite group, $x$ be a $p$-element of $G$, $B\in\mathcal{Bl}(G)$, and $\chi\in$ Irr$(B)$.
If $d_{\chi\mu}^x\neq 0$ and $\mu$ ...
- 1,755
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:
...
- 1,755
3
votes
1
answer
226
views
Can MAGMA compute almost projective $kG$-homomorphisms?
Let $G$ be a finite group and $k$ be a finite field (big enough) whith char$(k)=p$ and $p\mid |G|$.
Let $M$ be a finitely generated $kG$-module.
We denote the first syzygy of $M$ by $\Omega(M)$, i.e....
- 1,755
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 ...
- 1,755
4
votes
1
answer
436
views
Connectedness of the stabilizer in a semisimple group of a semisimple element in the Lie algebra: a reference request
Let $G$ be a (connected) semisimple algebraic group over an algebraically closed field $k$ of characteristic 0.
We consider the adjoint representation
$$ {\rm Ad}\colon G\to {\rm GL}({\mathfrak g}),$$
...
- 14.2k
5
votes
0
answers
266
views
Character tables of finite groups and isomorphism
I'd like to ask the following question:
Let $G$ and $H$ be finite groups.
Is there a useful criterion involving the ordinary character table which assures that $G$ and $H$ are isomorphic as groups?...
- 1,755
3
votes
0
answers
94
views
Clifford correspondence(s) from Fong-Reynolds theorem
The Fong-Reynolds theorem states a certain relationship between blocks of a normal subgroup $N\unlhd G$ and blocks of $G$, sometimes called the "Clifford correspondence for blocks". If one phrases ...
- 9,650
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 ...
9
votes
1
answer
460
views
Connections between linear representations and permutation representations
A finite group $\Gamma$ might be represented by a linear transformation
$$\rho : \Gamma\to\mathrm{GL}(\Bbb R^d),$$
or by permutations
$$\phi :\Gamma\to\mathrm{Sym}(n).$$
Of course, latter ones can ...
- 13.6k
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 ...
- 1,755
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 ...
- 111
4
votes
0
answers
160
views
What are the zonal spherical functions for a finite unitary group acting on a unit sphere?
Given a prime power $q$ and a dimension $d$, consider the Hermitian form $(\cdot,\cdot) \colon \mathbb{F}_{q^2}^d \times \mathbb{F}_{q^2}^d \to \mathbb{F}_{q^2}$ given by
$$
(x,y) = \sum_{i\in [d]} ...
- 7,633