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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, ...
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 ...
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\...
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 ...
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 ...
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 $G \leq \operatorname{GL}_n(K)$ ...
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 ...
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,...
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 ...
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 ...
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 ...
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 ...
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 ...
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 $d$, $...
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 ...
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^{\...
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 ...
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 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 ...
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 $G$...
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{...
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 ...
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 ...
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 ...
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}(...
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.), ...
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-...
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, ...
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 ...
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{...
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 $\rho_i:\Gamma\to SL(2,\mathbb{R})$ ...
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 ...
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 ...
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{\...
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\...
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}} ...
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. ...
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 ...
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 ...