All Questions
39 questions
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
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
33
votes
5
answers
4k
views
Is every (finite-dimensional, complex) representation of a finite group defined over the algebraic integers?
Is every (finite-dimensional, complex) representation of a finite group defined over the algebraic integers?
Apologies in advance if this is obvious.
Edit, 5/31/24: Since this question is getting some ...
- 118k
27
votes
5
answers
3k
views
Are there any nontrivial ring homomorphisms $M_{n+1}(R)\rightarrow M_n(R)$?
Let $R$ be a finitely generated ring with identity, $M_n(R)$ the set of $n\times n$ matrices. Are there any nontrivial ring homomorphisms $M_{n+1}(R)\rightarrow M_n(R)$? This should be an elementary ...
- 1,373
13
votes
1
answer
358
views
Cartography of the duals of GL, PGL, SL, etc
A short version of this question could be
What are the duals of $PGL(2,\mathbf{Q}_p)$, $PGL(2,\mathbf{R})$ and $PGL(2,\mathbf{C})$?
I should obviously add some precisions.
there are different ...
- 5,424
12
votes
1
answer
1k
views
What are the $p$-adic representations of $\hat{\mathbb{Z}}$ ?
A continuous representation $\hat{\mathbb{Z}} \rightarrow GL_n(\mathbb{Q}_p)$ is determined by the image of $1$. But the image of $1$ does not always defines such a representation (consider for ...
12
votes
2
answers
903
views
Which compact groups have nonisomorphic irreducible representations of the same dimension?
If $\Gamma$ is a compact simply-connected semisimple Lie group, then the Weyl Dimension Formula tells us exactly which dimensions it can act irreducibly on.
For certain $\Gamma$, it is easy to find ...
- 5,191
11
votes
0
answers
359
views
Representation theory of $\operatorname{GL}_2(\mathbb Z/n\mathbb Z)$
Is there a nice reference for the finite dimensional (characteristic 0) representation theory of $\operatorname{GL}_2(\mathbb Z/n\mathbb Z)$ and $\operatorname{PGL}_2(\mathbb Z/n\mathbb Z)$ for ...
- 7,746
8
votes
0
answers
366
views
Higher-dimensional generalization of Pink's theorem
Pink's theorem in the title of the question refers to the main theorem of Pink's paper "Compact Subgroups of Linear Algebraic Groups" that appeared in Journal of Algebra (206) in 1998. It essentially ...
- 26k
7
votes
2
answers
346
views
When is Ad(pi) an irreducible representation ?
For a finite group $G$ with a representation $\pi:G\to GL_n(\mathbb C)$ one can define the adjoint representation $Ad$ as the non-trivial summand in $End(\pi)$, i.e.
$End(\pi)=\pi\otimes \pi^{\vee}=1\...
- 71
7
votes
1
answer
456
views
Can Galois conjugates of lattices in SL(2,R) be discrete?
Let $\Gamma$ be a lattice in $SL(2,\mathbb{R})$. Suppose that the trace field of $\Gamma$ is a totally real number field of degree $d$. This gives $d$ homomorphisms $\rho_i:\Gamma\to SL(2,\mathbb{R})$ ...
- 454
7
votes
0
answers
261
views
Invariant lattices of group representations over a $p$-adic field
Let $G$ be a finite group, $K$ be a $p$-adic field with an uniformizer $\pi$ and residue field $k \cong \Bbb F_q$, and $V$ be an irreducible representation of $G$ over $K$.
Let $X_{V}^G$ be the set ...
- 6,622
6
votes
3
answers
1k
views
An application of Maschke's theorem
I've been teaching some elementary representation theory to undergraduates, and want to provide applications of Maschke's theorem to complex group algebras to present in class. In particular, I'd like ...
- 1,472
5
votes
1
answer
231
views
Subgroups of algebraic groups
Is anyone aware of a result (or a counterexample) along the following lines: let $G$ be an algebraic group over $\mathbf Z$. Let $H$ be a finite group such that $H$ occurs as a subgroup of $G(\bar{\...
- 53
5
votes
2
answers
367
views
Proving properties of metaplectic groups without using explicit cocycle
I learned the metaplectic group from the book of Gelbart, Weil's representation and the spectrum of the metaplectic group. It seems to me that most of the properties of the metaplectic group are ...
- 3,432
5
votes
1
answer
251
views
Quotient of Projective line over rationals with an infinite subgroup of PGL(2,Q)
I am looking for references for the following; how to calculate quotient of the projective line over the field of rationals with an infinite subgroup of PGL(2,Q), e.g, of the form
$
\left(
\begin{...
- 641
5
votes
1
answer
465
views
The induced representations of $SL(2, F)$.
Let $G=SL(2,F)$ and $I=J_{0}\cap J_{1}$ be the Iwahori subgroup of $SL(2, F)$, where $J_{0}=\left(
\begin{array}{cc}
\mathcal{O}_{\mathbb{F}} & \mathcal{O}_{\mathbb{F}} ...
5
votes
0
answers
97
views
Is there a composite-order generalization of the homomorphism on Rep(Z/p) giving total dimension of Tate cohomology?
Let $p$ be a prime, let $\mathbb{Z}_p$ be the ring of $p$-adic integers, and let $G$ be a cyclic group of order $p$. It is rather well-known that finite rank $\mathbb{Z}_p$-free representations of $G$...
- 45.7k
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
4
answers
1k
views
Are there some nontrivial group homomorphisms from $SL_n(\mathbb{Z})$ to $GL_{n-1}(\mathbb{Z})$ for $n\geq3$?
Are there some nontrivial group homomorphisms from $SL_n(\mathbb{Z})$ to $GL_{n-1}(\mathbb{Z})$ for $n\geq3$ except the determinant? This should be a natural question and any references are welcomed.
...
- 1,373
4
votes
2
answers
367
views
An infinite profinite group such that any $p$-adic representation has finite image
Fix a prime $ p $. We call an infinite profinite group $ G $ a Fontaine-Mazur group (with respect to $ p $) if every continuous homomorphism $ G\to {\rm GL}_n(\overline{\mathbb{Q}}_p) $ has finite ...
- 863
4
votes
2
answers
170
views
Action of certain endomorphisms on Pontriyagin dual
Let $R$ be a finite ring and $F$ be an algebraically closed field in which $|R|$ is invertible. Does there exists an $F$-valued character $\chi$ of $(R, +)$ such that every character $\psi$ is of the ...
- 43
4
votes
1
answer
403
views
A question on bi-character of finite abelian group
Setting: $G$ is a finite abelian group and any bicharacter on $G$, where a bi-character on $G$ is a map $b:G \times G \to \mathbb{Q}/\mathbb{Z}$ such that $$b(x+y,z)=b(x,z)+b(y,z),b(x,z+y)=b(x,z)+b(x,...
- 41
4
votes
1
answer
336
views
Infinite, finitely generated linear group has indices of subgroups divisible by infinitely many primes
This question was previously asked at Math.SE, but didn't receive much attention.
Let $G$ be an infinite finitely generated group and suppose that $G$ is linear, say $G \leq \operatorname{GL}_n(K)$ ...
- 2,821
4
votes
0
answers
176
views
Smooth admissible representations, Hom, tensor and extension of scalars
(Remark: This has previously been posted on math.stackexchange, but I believe it might be suitable for this site as well. https://math.stackexchange.com/questions/1428350/smooth-admissible-...
- 203
3
votes
1
answer
395
views
Waldspurger Formula as a Torus Integral
I have a research-level but not necessarily new question about certain equidistribution problems. If $\phi \in L^2(S^2)$ then we could define the Weyl sums:
$$ \int \phi \, \mu_d = \frac{1}{|\mathcal{...
- 22.8k
3
votes
0
answers
171
views
Conjugacy in metaplectic groups
Let $F$ be a non-Archimedean local field (characteristic 0) and $G=GL(2,F)$. Let $\tilde{G}$ be "the" metaplectic double cover of $G$ (defined using an explicit cocycle as in Gelbart's book (Weil's ...
- 185
3
votes
0
answers
130
views
Natural permutation representations of the orthogonal group of a lattice
Let $A\in M_n(\Bbb Z)$ be a positive definite matrix so it gives a metric on $\Bbb Z^n$. Let $G=O(q,\Bbb Z^n)$ be the subgroup of $GL_n(\Bbb Z)$ preserving the metric. For every positive integer $d$, $...
- 6,622
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, ...
- 14.1k
2
votes
1
answer
185
views
Fields of definition of parabolically induced representations of $\mathrm{SL}(2,q)$
Let $\alpha_0$ be the unique non-trivial character satisfying $\alpha_0^2=1$ of the split torus $\mathrm{T} \subset \mathrm{SL}(2,q)$ and denote by $\mathrm{R}(\alpha_0)$ the character of $\mathrm{SL}(...
- 381
2
votes
0
answers
228
views
Satake correspondence for groups over finite field
I asked the same question in MSE, but I didn't get any answer. So I decided to post it here, too.
In Langlands' program, Satake correspondence gives a correspondence between unramified ...
- 2,215
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 ...
- 2,429
1
vote
1
answer
312
views
On the number of structure of $F_p[G]$-modules
Let $A$ be an abelian group and $G$ be a group. A short exact sequence of
groups like $1\longrightarrow A\longrightarrow E\longrightarrow
G\longrightarrow 1$ is called an extension. We say that $E$ is ...
- 463
1
vote
1
answer
232
views
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\...
- 14.1k
1
vote
1
answer
285
views
centralizer of a n-cyclic permutation matrix over F_2 in GL(n,2)
This is a continuation of this question, where I talked about the case $n=2^k$. Let $C$ be the $n\times n$-permutation matrix over $\mathbb{F}_2$ of the $n$-cycle. We needed to know the explicit ...
- 14k
1
vote
0
answers
174
views
What are the irreps in this canonical action of $\operatorname{PGL}_2(F_q)$?
Consider the permutation action of $\operatorname{PGL}_2(\mathbb F_q)$ on $\mathbb P^1(\mathbb F_q)$ by fractional linear transformations. We can consider the associated (complex) representation of ...
- 7,746
1
vote
0
answers
77
views
Existence of a certain direct summand
Let $P$ be a cyclic group of order $p^n$. Let $M$ be a $\mathbb{Z}_p[P]$-module and $M_p :=\mathbb{Q}_p\otimes M$ be the associated $\mathbb{Q}_p[P]$-module. When will $M_p$ have a direct summand ...
- 1,283
0
votes
1
answer
283
views
Is any abelian subgroup of a semidirect product isomorphic to a direct product of abelian subgroups? [closed]
Let $H$ and $K$ be groups and $V$ an abelian subgroup of the semidirect
product $\ H\rtimes K$. Do there exist abelian subgroups $H^{\prime }\leq H$
\ and $K^{\prime }\leq K$ \ such that $V\cong H^{\...
- 463
0
votes
0
answers
65
views
Higher-order obstructions in thin group orbits
Let $G$ be a finitely generated group acting on the integers $\mathbb{Z}$. Let $O_a = \{g \cdot a : g \in G\}$ be the orbit of an integer $a$ under this action. Assume that $O_a$ is a thin orbit, ...
