All Questions
Tagged with galois-theory rt.representation-theory
18 questions
4
votes
0
answers
180
views
Subgroups that conjugate-cover the ambient group
Let $G$ be a finite group, and suppose that a set of proper subgroups $H_1,\dotsc,H_n$ satisfy $G=\bigcup_{g\in G}\bigcup_{i=1}^nH_i^g$, where $H_i^g$ is the conjugate of $H_i$ by $g$. In this case, ...
3
votes
1
answer
459
views
Why are smooth irreducible representations of the Weil group finite dimensional?
I am trying to understand the proof in Bushnell and Henniart - The local Langlands conjecture for $\operatorname{GL}_2$ of the fact (§28.6, Lemma 1) that smooth irreducible representations of the Weil ...
- 12.9k
2
votes
0
answers
125
views
Semisimplicity of induced representation of a irreducible representation
This question occurs when I read this one.
Suppose $k$ is a field of char $0$ (not necessarily complex numbers, in my case it's $\mathbb Q_p$), and $G$ is a profinite group (in my case, $\operatorname{...
- 775
6
votes
1
answer
513
views
Trying to understand the topology of the Weil group for the local Langlands conjecture
I am trying to study the representation of the Weil group from the book "The Local Langlands Conjecture for $GL(2)$". I have some problem with the topology of this group.
Let $F$ be a non ...
- 28.7k
3
votes
1
answer
296
views
Distinct characters with the same character values, outer automorphisms and Galois conjugation
Given an (irreducible complex) character of a finite group the following three construction all yield another irreducible character of the same degree:
multiplying by a degree 1 character
applying an ...
- 148k
2
votes
1
answer
193
views
Irreducible components of a cyclic extension over $ \mathbb{Q} $
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Gal{Gal}$Let $ L $ be a cyclic Galois extension of $ \mathbb{Q} $ of degree $ 6 $. So $ G = \Gal(L/\mathbb{Q}) $ is a cyclic group of order $ 6 $. Then ...
- 12.9k
6
votes
0
answers
179
views
Characteristic polynomials of Cartan matrices of Lie algebras
Let $L$ be a simple Lie algebra over $\mathbb{C}$ with Cartan matrix $C_L$ (as in https://de.wikipedia.org/wiki/Cartan-Matrix )
Question 1: Is is true that the characteristic polynomial $f$ of $C_L$ ...
- 26.5k
2
votes
0
answers
549
views
Fontaine - Wintenberger field of norms and imperfect case
Let $K$ be a complete discrete valued field whose residue field $k_K$ has characteristic $p$ and has the property that $[k_K:k_K^p]=p^d$ for some $d$. Let $t_{\alpha}, 1 \leq \alpha \leq d$ be a set ...
- 313
1
vote
0
answers
59
views
Weyl theorem for non specified primitive root of unity
Let $\omega=e^{2i \pi/p}$.
Weyl theorems give all representations of matrix algebra span by $A,B$ such that either
$AB=\omega BA, A^p=B^p=I$,
or
$(k,l)\mapsto A^kB^l$ is a irreducible ...
- 583
3
votes
0
answers
129
views
Galois descent for profinite groups acting on local fields
Suppose that $G$ is a profinite group acting faithfully on a field $L$ and assume moreover that the action is admissible, that is, that every $l \in L$ is stabilized by an open subgroup of $G$. If ...
- 2,197
2
votes
0
answers
138
views
Local polynomials of Frobenius-semisimple Weil representations which are tensor products of an Artin representation and an unramified character
Let $K$ be a local field and $\rho: W_K \to \operatorname{GL}(V)$ be a Weil representation. The for any finite extension $F/K$, we define the local polynomial
$$
P(\rho|_F,T) = \det{(1 - \operatorname{...
- 103
1
vote
0
answers
105
views
Local factors determine Weil representations - proof of the Artin representation case
This post can be seen as a continuation of this post I created on MathOverflow.
I want to understand the proof of the following Theorem from "Euler Factors determine Weil Representations" by Tim and ...
- 103
2
votes
1
answer
317
views
Local factors determine Weil representations - proof of the cyclic case
I already created this post on Math Stack Exchange but I was not so sure if this question fits better here. If it is not, I want to apologize in advance and feel free to delete my post.
I want to ...
- 13k
18
votes
1
answer
435
views
Splitting the Resultant, as when the Determinant becomes the square of the Pfaffian
The Determinant of an $n\times n$ matrix, viewed as a polynomial in the entries, is irreducible. But when it is restricted to the subspace of alternate matrices, it becomes reducible, actually the ...
- 52.3k
14
votes
1
answer
790
views
Noether-Deuring for injections and surjections?
Noether-Deuring theorem (not in the strongest form, but in the one I usually need):
Let $L\diagup K$ be a field extension. Let $A$ be a $K$-algebra which is finite-dimensional as a vector space over $...
- 33.8k
7
votes
1
answer
1k
views
Artin representations in Serre's book 'local fields'
Let $K$ be a complete local field with discrete valuation, and let $L/K$ be a finite Galois extension. Use $G=Gal(L/K)$ to denote the Galois group.
In Serre's book 'local fields', chapter 6, a ...
- 18.7k
1
vote
2
answers
416
views
sheaves of representations on galois groups, can there be interesting cohomology?
Consider a field $K$ (of characteristic 0, say) and its absolute galois group $G_K^{ab} = Gal(\overline{K}/K)$, given the Krull topology: $U_E(\sigma) = \sigma Gal(\overline{K}/E)$ form a basis of the ...
- 5,670
4
votes
2
answers
1k
views
Galois theory of endomorphism rings of irreducible representations
Let $k$ be a field which I don't suppose to be algebraically closed. Then, the endomorphism ring of any irreducible representation of a finite group over $k$ is a division ring.
What is known about ...
- 3,246