All Questions
525 questions
3
votes
0
answers
65
views
p-torsion in the Tate-Shafarevich group of supersingular elliptic curves
Let $E$ be a supersingular elliptic curve over $\mathbb{F}_p(t)$. Is something known on the $p$-torsion of the Tate–Shafarevich group in this case? In particular, I would like to know if (or if known ...
- 103
1
vote
0
answers
89
views
Generic reducedness of geometric generic fibre
Let $f:X\to Y$ be a surjective morphism between two projective schemes over a field of characteristic $p>0$. Also assume that $X$ is smooth,$Y$ smooth & irreducible and $f_*\mathcal{O}_X=\...
- 5,986
12
votes
0
answers
347
views
Does every finite group have a small projective representation (over some ring)?
Question. Let $G$ be a finite group. Can we find some (commutative) ring $R$ and some positive integer $d=O(\log\lvert G\rvert)$ such that $G$ can be found as a subgroup of $\operatorname{PGL}_d(R)$?
...
- 602
3
votes
0
answers
91
views
Are the reductions of the cuspidal characters of GL2(Fq) distinct?
Let $p$ be an odd prime and $q=p^n$ for some $n \geq 1$. If $\mathbb{F}_q$ is the unique, up to isomorphism, finite field with $q$ elements then the cuspidal representations of the group $\rm{GL}_2(\...
- 117
0
votes
0
answers
190
views
About Chern classes via Atiyah class
I am trying to understand a construction of the Chern classes of a vector bundle $\mathcal{E}$ via the Atiyah class, like is done in this text and here in section 1.4. I am interested in the case ...
- 348
18
votes
2
answers
888
views
Serre’s comment on Hurwitz: connecting FLT to points of finite order on elliptic curves
In the paper Sur les représentations modulaire de degré 2 de Gal$(\overline{\mathbb{Q}}/\mathbb{Q})$, Serre makes the following comment:
Remarque. La relation existant entre "solutions de l'...
- 6,734
8
votes
1
answer
705
views
When does a finite group have finitely many indecomposable representations?
Let $G$ be a finite group and let $k=\mathbb F_p$. Then it is well-known that $G$ has finitely many irreducible modules.
However, in general $G$ does not have finitely many indecomposable ...
- 3,054
1
vote
0
answers
86
views
Unitary representations of the symmetric group over finite fields
I am interested in understanding the unitary representations of the symmetric group over $\mathbb{F}_{q^2}$. In general, some comments here are relevant
Unitary representations of finite groups over ...
- 571
1
vote
0
answers
82
views
Behavior of translation functors in characteristic $p$
Let $G$ be a semisimple and simply connected algebraic group over an algebraically closed field of characteristic $p>0$, and let $\mathfrak g$ be the Lie algebra of $G$. Let $U_\chi(\mathfrak g)$ ...
- 2,974
3
votes
0
answers
192
views
How can I prove this stronger version of Fedder's Criterion?
I was reading Fedder's original paper which proved what is now known as "Fedder's criterion". I noticed that the abstract stated something which is a priori stronger than what is proved in ...
- 317
5
votes
1
answer
344
views
Surjection onto endomorphisms of multiplicative group of a field
Let $k$ be an algebraically closed field of characteristic $p > 0$. Denote by $k^\times$ the multiplicative group of $k$. There is a ring homomorphism given by restriction to $k^\times$
$$
\mathbb{...
- 53
4
votes
1
answer
156
views
Question about defect subgroups
Let $G$ be a finite group over an algebraically closed field $\Bbbk$ of characteristic $p>0$. Let $b$ be a block of $G$. Then the a defect subgroup $D$ of $b$ is one for which every module in $b$ ...
- 1,077
18
votes
2
answers
1k
views
The mysterious significance of local subgroups in finite group theory
EDIT 21/12: Even if there are no conclusive answers to these questions, I would very much like to know if anyone has noted and attempted to explain the mysterious significance of local subgroups: are ...
- 1,433
17
votes
2
answers
2k
views
How to think of algebraic geometry in characteristic p?
How does a working mathematician usually think about algebraic geometry in characteristic $p$? For the sake of concreteness, and to make things more "geometric" (whatever that means), let's ...
5
votes
1
answer
389
views
What goes wrong with the Brauer construction for a module over a complete DVR?
Let $G$ be a finite group, $\mathcal{O}$ a complete discrete valuation ring, and $F$ the residue field. The Brauer construction for a fintely-generated $\mathcal{O}G$-module $M$ with respect to a $p$-...
- 75
1
vote
2
answers
197
views
What are the finite étale coverings of a quasi-hyperelliptic surface?
Let $X$ be a quasi-hyperelliptic surface in characteristic 3 where the canonical bundle $K_X$ is trivial.
Question: Is there a finite étale covering $Y \rightarrow X$ such that
$Y$ is an abelian ...
- 9,501
2
votes
1
answer
176
views
Isomorphisms $S^d(S^m(V)^*) \cong \Lambda^d(S^{m+d-1}(V)^*)$
$\DeclareMathOperator\SL{SL}$Let $K$ be an algebraically closed field. Let $V$ be a 2-dimensional vector space over $K$. The group $G=\SL_2(K)$ acts naturally on $V$ by left-multiplication on column ...
- 185
4
votes
0
answers
129
views
How does one compute the group action of the automorphism group on integral cohomology?
Suppose I have a curve $X$ (for concreteness, we can take $X$ to be a smooth, projective curve over a finite field $\mathbb F_q$, and even more concretely consider the family of curves described by ...
- 7,746
5
votes
2
answers
473
views
Fixed points of a linear abelian p-group in characteristic p
Let $V=\bigoplus_{n\geq1}\mathbb F_{p}\cdot e_{n}$ be an $\mathbb F_{p}$-vector space of countable dimension, and write $V_{n}=\operatorname{Vect}(e_{1},\dotsc,e_{n})$. Let $G$ be a (possibly infinite)...
- 105
7
votes
1
answer
348
views
Do you know a survey of modular Lie algebras and its representations?
When I was an university student, I liked reading some books about the representation theory of finite groups or Lie algebras and I was interested in explicit constructions of irreducible ...
- 71
2
votes
1
answer
106
views
Does a polynomial $P(X,Y)$ that specializes to a polynomial $P(x_0,Y)$ with distinct roots in $\overline{k}$ have distincts roots in $\overline{k(X)}$
Let $k$ be a field, $P\in k[X][Y]$ be a monic polynomial of degree $n$ in $Y$.
I am looking for a simple proof of the following fact.
"If there exists $x_0\in k$ such that $P(x_0,Y) \in k[Y]$ has ...
- 2,521
0
votes
0
answers
149
views
Is there a way to find the eigenvalues of a matrix using character table?
I am studying applications of representation theory. I want to know if there is a procedure to find the eigenvalues and eigenvectors of a matrix using the character table of the Group acting on its ...
- 1
0
votes
0
answers
42
views
Which finite group schemes over positive characteristic have projectivity detected by a set of (generalized) cyclic subgroups?
A finite algebra $B$ over a field $K$ has projectivity detected by a set $S = \{f_i : A \to B \mid i \in I\}$ of maps into $B$ from a fixed algebra $A$ if for any finite module $M$, $M$ is projective ...
- 542
10
votes
0
answers
371
views
How large must the characteristic of $k$ be, for the cohomology of the Lie algebra $\mathfrak{sl}_n(k)$ to be exterior as in characteristic zero?
$\DeclareMathOperator\SU{SU}$In this question, all Lie algebra cohomology is of the form $H^*(\mathfrak{g}; k)$, with $k$ the trivial one-dimensional representation of $\mathfrak{g}$. All Lie algebra ...
- 1,335
4
votes
0
answers
135
views
Nilpotent orbits in characteristic $0$ vs. positive characteristics
Let $G_\mathbb{C}$ be a connected reductive group over $\mathbb{C}$ with Lie algebra $\mathfrak{g}_{\mathbb{C}}$. For any algebraically closed field $k$, let $G_k$ denote the connected reductive group ...
- 2,751
4
votes
0
answers
108
views
Shafarevich conjecture for Abelian varieties over global function fields
Let $S$ be a finite set of places of a global function field $K$. Are there finitely many Abelian varieties over $K$ with good reduction outside $S$? What if we exclude isotrivial families?
- 679
11
votes
1
answer
381
views
Chromatic representation theory of the symmetric groups?
We know that in characteristic 0, the group ring of the symmetric group $\Sigma_n$ splits via one idempotent for each partition of $n$.
In characteristic $p$, I believe the analogous statement is that ...
- 64k
3
votes
0
answers
192
views
Congruences between Eisenstein series and cusp forms
Let $k\geq 4$ be an even integer. Let $p>k$ be a prime
such that $p\mid B_k$, the $k$th Bernoulli number. Then there is a primitive cusp form $f=\sum_{n\geq1}c(n, f)q^n$
of weight $k$ and level $1$ ...
- 347
4
votes
0
answers
183
views
Characters of finite fields - Reference request
Let $\mathbf{F}_q$ ($q=p^f$) be a finite field. We are interested in the characters $\chi: \mathbf{F}_q\rightarrow \mathbf{K}$ ($\chi(0)=0$) where the $ \mathbf{K}$ is an alg.closed field of ...
- 41
3
votes
1
answer
221
views
Asymptotics for number of $p$-regular partitions of $n$
The number of simple modules $D^{\lambda}=S^{\lambda}/S^{\lambda}\cap (S^{\lambda})^{\bot}$ of the symmetric group over a field $k$ such that $\text{char}(k)=p > 0$ is the number of $p$-regular ...
- 571
5
votes
1
answer
265
views
Schur functors = Weyl functors in characteristic zero?
I asked this question on Math Stack Exchange https://math.stackexchange.com/questions/4789924/schur-functors-weyl-functors-in-characteristic-zero, but I got no answers, so I ask the same question here....
- 137
3
votes
0
answers
109
views
Exact structures on representations of a finite group
For simplicity assume $G$ is a (finite) $p$-group, and $k$ is field of characteristic $p$, so that there exists a unique simple $kG$-module the trivial module $k$. I am looking for a class of short ...
- 542
2
votes
1
answer
174
views
Using the mapping cone to show that a chain map defines a stable equivalence between two symmetric algebras
This question is about an argument in the proof of Theorem 9.8.8 in Linckelmanns Block Theory of Finite Group Algebras. I need to understand the argument in order to do something similar in my ...
- 75
5
votes
0
answers
210
views
Can modular representation theory be used to prove Sylow's existence theorem?
Edit 20/12: I added a more precise question at the bottom of the post.
Given a finite group $G$ and a prime $p$, we want to prove that $G$ has a $p$-subgroup $P$ such that $|G:P|$ is not divisible by $...
- 1,433
1
vote
1
answer
194
views
Non-split extension of representations of $\mathrm{GL}_2$ and $\mathrm{Hom}$
Let $0\to V_1\to V\to V_1\to0$ be a sequence of representations of $\mathrm{GL}_2(\mathbb{Q}_p)$ over $\overline{\mathbb{F}}_p$, where $V_1$ is irreducible, smooth and admissible. Assume that this ...
user515481
6
votes
0
answers
266
views
What is the Fourier transform in modular representation theory?
For a finite group $G$ there is the Fourier transform $\displaystyle \hat{f}(\rho)=\sum_{g \in G} f(g)\rho(g)$ with inverse $$\displaystyle f(g)=\frac{1}{|G|}\sum_{\rho}d_{\rho}\operatorname{Tr}\left(\...
- 571
4
votes
0
answers
313
views
What is $\dim D^{\lambda}$ for the symmetric group?
What are the dimensions of the simple modules $D^{\lambda}=S^{\lambda}/S^{\lambda}\cap (S^{\lambda})^{\perp}$ for the modular representation theory of $S_n$, i.e. $\operatorname{char}(k)=p>0$?
I ...
- 571
5
votes
1
answer
160
views
Derived subalgebra of a restricted Lie algebra
Let $L$ be a restricted Lie algebra over a field of characteristic $p>0$. It is well known that the commutator subalgebra $[L,L]$ is not necessarily restricted (that is, closed under the $p$-map).
...
- 630
2
votes
0
answers
169
views
$\mathrm{Ext}^i(\pi_1, \pi_2)\neq0$ implies same central character
If $\pi_1$ and $\pi_2$ are two smooth admissible representations of $\operatorname{GL}_2(\mathbb{Q}_p)$ over $\overline{\mathbb{F}}_p$ with central characters. I want to prove that if $\pi_1$ has ...
user515481
2
votes
0
answers
127
views
Classification of restricted Lie algebras of reductive groups
$\DeclareMathOperator\Lie{Lie}$Let $G/K$ be a reductive group over a field $K$. In characteristic $0$ the Lie algebra is invariant under base change of fields, so to understand $\Lie(G)$ it is enough ...
- 461
24
votes
0
answers
814
views
Revising the proof of CFSG
This is an oft-quoted excerpt from John Thompson's article "Finite Non-Solvable Groups":
“... the classification of finite simple groups is an exercise in taxonomy. This is
obvious to the ...
- 1,433
20
votes
0
answers
408
views
Ado's theorem and the reduction to positive characteristic
The synopsis: proofs of Ado theorem in positive characteristic are simple, and in characteristic $0$ are difficult. Can one infer the characteristic $0$ case from the positive characteristic case?
The ...
- 2,934
79
votes
12
answers
13k
views
Is there a high-concept explanation for why characteristic 2 is special?
The structure of the multiplicative groups of $\mathbb{Z}/p\mathbb{Z}$ or of $\mathbb{Z}_p$ is the same for odd primes, but not for $2.$ Quadratic reciprocity has a uniform statement for odd primes, ...
- 118k
2
votes
1
answer
170
views
Automorphism of positive characteristic field
Suppose $K$ is a field with $\text{char}(K) \geq 0$. Let $L$ be a cyclic extension of $K$ with degree $2n$. We consider the generator $\sigma$ of the Galois group $\text{Gal}(L/K)$.
I am interested in ...
- 923
8
votes
3
answers
2k
views
Representation theory of $\text{SL}(2,\mathbb{Z})$
The group $\text{SL}(2,\mathbb{Z})$ is the group of two-by-two matrices with integer entries and determinant one. This is a very simple definition. Yet its representation theory seems quite wild to me ...
- 231
2
votes
1
answer
160
views
Finding the mistake in an argument concerning $F$-finite $F$-split local Cohen--Macaulay ring of dimension $1$
Let $R$ be a commutative Noetherian ring, and $\phi: R \to R$ be a ring homomorphism. For an $R$-module $M$, let $^{\phi}M$ be the $R$-module defined via restriction of scalars via $\phi$, i.e., as ...
- 412
6
votes
1
answer
771
views
A regular, geometrically reduced but non-smooth curve
Can anyone give an example of a projective, regular, geometrically reduced but non-smooth curve ?
Of course, the base field should be imperfect.
In Exercise 4.3.22 of Qing Liu's book Algebraic ...
- 620
9
votes
2
answers
783
views
Can we use formal groups to recover Lie-theoretic representation theory in characteristic p?
In differential geometry, Lie's theorems allow us to integrate any Lie algebra representation to a Lie group representation. The algebraic version of this is more complicated (and I'm not terribly ...
- 1,969
4
votes
1
answer
248
views
Relation between positive roots of $E_8$ and $\mathbb{F}_2^8 \setminus \{0\}$
There exists an explicit bijection (due to Cayley, that has built up a very nice table to describe this) between the positive roots of the lattice $E_7$ and $\mathbb{F}_2^6 \setminus \{0\}$ (where $\...
- 3,779
6
votes
1
answer
141
views
When is the group algebra a product of local rings up to Morita equivalence?
Let $KG$ be the group algebra of a finite group $G$ over a field of characteristic $p$.
Question 1: Is there a characterisation when $KG$ is Morita equivalent to a product of local rings?
This ...
- 26.5k