All Questions
35 questions with no upvoted or accepted answers
9
votes
0
answers
254
views
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
8
votes
0
answers
545
views
What are the character tables of the finite unitary groups?
I need to know the (complex) character table of the finite unitary group $U_n(q)$. Lusztig and Srinivasan (1977) provide an abstract description, but parsing it requires a stronger background in ...
- 7,633
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
6
votes
0
answers
225
views
Parshin's buildings for higher local fields
What is the status of the theory of buildings for higher local fields?
I know that there are some papers of Parshin, in which he describes some examples, like $PGL_2$ and $PGL_3$ over two-...
- 17.4k
5
votes
0
answers
140
views
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
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
5
votes
0
answers
219
views
Character tables of the p-core of the binary modular congruence group of p-power level
Let $p \geq 5$ be a prime and let $n$ be positive integer. In his Ph.D thesis (See The characters of binary modular congruence group, Bulletin of the
American Mathematical Society. 79 (1973), no. 4.), ...
- 2,396
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
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 ...
- 286
4
votes
0
answers
176
views
Is there a notion of "tame" representations of $GL_n(Z)$?
This is a followup to this question about the (left) noetherianity of the group ring of $GL_n(\mathbf{Z})$:
Does GL_n(Z) have a noetherian group ring?
Given that $\mathbf{Z}[GL_n(\mathbf{Z})]$ is ...
- 10.7k
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
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
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
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
3
votes
0
answers
184
views
Mackey Obstruction Class with Integral Coefficients
Consider an exact sequence of groups
\begin{equation}
1\rightarrow H\rightarrow K\rightarrow G \rightarrow1~.
\end{equation}
Mackey theory enables us to understand representations of $K$ in terms of ...
- 2,097
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
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
0
answers
209
views
What is known about 2-modular representations of Ree groups of type $F_4$?
A too-vaguely worded question posted today about Suzuki and Ree groups reminds me to revisit a concern I never followed up years ago when assembling information about modular representations of finite ...
- 52.9k
3
votes
0
answers
264
views
How do I determine the smallest dimension of an irreducible $\mathbb{F}_p[G]$-module with a prescribed trivial fixed point space?
This is a crosspost from MSE since I haven't found an answer there yet.
I am not very familiar with modular representation theory or Brauer theory yet, however lately I have needed to use $\mathbb{F}...
- 1,173
2
votes
0
answers
118
views
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)$$
where $T(3)$ is the translation group.
I am looking for a reference classifying all the finite-...
- 31
2
votes
0
answers
137
views
$p$-adic Banach group algebra
Let $G$ be a discrete group. Consider the Banach $\mathbb{Z}_p$-algebra: $$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
2
votes
0
answers
88
views
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
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
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
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
2
votes
0
answers
414
views
Mixed up by definitions of mildly mixing
Here are two setup where the notion of "mildly mixing" comes up: for representations and for group acting by measure preserving transformations (see definitions below). Since a natural class of ...
- 4,432
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)....
- 267
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 ...
- 631
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
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
1
vote
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
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
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
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
0
votes
0
answers
289
views
Modular representations of the symplectic group
Let G=Sp(2m,2) be a finite symplectic group acting on $F_2^{2m}$. This group G acts 2-transitively on $\Omega_{+}$ and on $\Omega_{-}$. Let $F$ be an algebraic closure of $F_2$.
I am interested to ...
- 2,179