# Questions tagged [modular-representation-theory]

For questions about modular representation theory, the study of representations over a field of positive characteristic.

142
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### 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 $b$ ...

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### 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 ...

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### 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 ...

5
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### 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 $p$-...

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### 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 ...

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### 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 $1$ ...

7
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### 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 ...

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### 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)...

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### 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 ...

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### 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 $...

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### 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 ...

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### 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 ...

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### 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 ...

3
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### 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 ...

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### 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(\...

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### 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....

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### $\mathrm{Ext}^i(\pi_1, \pi_2)\neq0$ 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 ...

1
vote

1
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### 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 ...

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### $p$-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 ...

24
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### 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 ...

3
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0
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### Has the positivity part in Lusztig's conjecture been solved?

In Lusztig's paper "BASE IN EQUIVARIANT K- THEORY Ⅱ", He stated a conjecture about the positivity of coefficients in his famous conjecture.
Precisely, (In 9.20) He conjectured that
$\tilde{\...

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### Modular invariants of special linear groups acting on exterior powers

In general it can be very difficult to compute the invariants of a group acting on a module which is not a vector space over an infinite field. I am interested in the following example, motivated by ...

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### Context for Wiles defect criterion and patching

This is not a homework or a project question, just me trying to get acquainted to the subject: I am an MSc student who recently came across the Wiles defect numerical criterion (see, for example, ...

6
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### 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 ...

1
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0
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### Pure, residual, 2-dimensional, semisimple $G_{\mathbb{Q}}$ representations

Let $G_{\mathbb{Q}}$ be the absolute Galois group of the rationals. I want to know how many isomorphism classes of 2-dimensional, irreducible, pure Galois representations are there over a finite field ...

2
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### How to compute the character of the Steinberg module for the group $\mathrm{SL}_n$ over a field of characteristic $p$?

It is known that the Steinberg representation $V$ of the group $\mathrm{SL}_n$ over a field $k$ of characteristic $p$ (maybe one needs to assume that $k$ is perfect, I am not sure) is the irreducible ...

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### Characters of simple $\mathfrak{sl}_n$-modules in positive characteristic with subregular nilpotent central character

Consider representations of $\mathfrak{sl}_n$ in positive characteristic with a subregular nilpotent central character $\chi$ (i.e. $\chi$ is a nilpotent matrix whose Jordan normal form has two blocks ...

3
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### An analog of a BGG resolution in subregular case in positive characteristic

Consider representations of $\mathfrak{sl}_n$ in positive characteristic with a subregular nilpotent central character $\chi$. For every regular weight $\lambda$ of $\mathfrak{sl}_n$, we have the ...

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### Dual Coxeter numbers, Langlands dual groups, black holes and twisted compactification of 6d (2,0) A D E theories on a circle

A 6-dimensional (2,0) superconformal quantum field theory comes in Lie algebra A, D, E types. These theories do not have classical Lagrangian and are purely quantum.These theories on a torus ...

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### Frobenius kernel by Greenberg functor

I was not able to find much on representation theory (with a geometric perspective) of algebraic groups over local artinian rings. In particular, studying on the classic book of Jantzen, I was asking ...

2
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### Brauer pairs associated to a normalizer subsystem in the fusion system of a block of a finite group

Let $G$ be a finite group and let $k$ be an algebraically closed field of positive characteristic $p$. Let $b$ be a block of $kG$ and let $(P,e)$ be a maximal $(G,b)$-Brauer pair. For every subgroup $...

9
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### 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 ...

5
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1
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### Examples of a group $G$ and an $F$-representation $V$ where $\cup:H^1(G,F)\otimes H^1(G,V)\to H^2(G,V)$ is injective

Let $G$ be a group and $F$ a field. I am particularly interested in the case where $G$ is a uniform lattice in a Lie group and $F=\mathbb{F}_2$, or in finite groups $G$ where
$\operatorname{char} F$ ...

6
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3
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### 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 ...

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### What do the indecomposable objects of the homotopy category of chain complexes look like?

I am trying to understand indecomposable objects in the homotopy category of chain complexes. Let $\mathcal{A}$ be an abelian category. Denote by $C(\mathcal{A})$ the category whose objects are chain ...

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### Indecomposable objects in iterated functor categories

Let $\mathcal{O}$ be a DVR with a uniformizer $\pi$ and a finite quotient field $\mathbb{F}_q=\mathcal{O}/\pi$. Write $\mathcal{O}_r = \mathcal{O}/(\pi^r)$. Fix now an integer $r\geq 2$. We define ...

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### Representation theory of Kac-Moody algebras in positive characteristic

I have been trying to learn about the representation theory of Kac-Moody algebras in positive characteristic. However, my usual reference for the subject (Infinite dimensional Lie algebras by Kac) ...

11
votes

2
answers

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### Reference on the Chern-Simons theory and WZW models for mathematicians

I would like to ask if there are any beginner friendly references for learning CS theory and WZW models. It seems that most mathematical texts on the subjects begin with convenient definitions that ...

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### The radical of $kG$-modules

$\DeclareMathOperator\Rad{Rad}$Let $k$ be a finite field of $p$ elements. Let $G$ be an elementary abelian p-group and $V$ a $kG$-module corresponding to the representation $\alpha:G\rightarrow \...

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### Stable homotopy type of $BG^{\wedge}_p$ in algebraic terms

In the mid 90's, Martino- Priddy proved that given two finite groups $G, H$, the following are equivalent:
$\mathbb{F}_p\mathrm{Inj}(P,G)\cong \mathbb{F}_p\mathrm{Inj}(P,H)$ as $\mathbb{F}_p\mathrm{...

4
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### Generalization of the second Brauer-Thrall conjecture for arbitrary Artin algebras

Let $k$ be a field with infinite cardinality and $A$ a finite dimensional $k$-Algebra. The second Brauer-Thrall conjecture states the following: There are infinitely many natural numbers $n_1<n_2&...

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### Ext groups of tensor products of Specht modules

$\newcommand{\sym}{\mathfrak{S}}$
$\DeclareMathOperator{\Ext}{Ext}$
Let $k$ be a field of characteristic $p > 0$ and $a < p \leq b$ so that the $k\sym_a$-modules are semisimple but $k\sym_b$-...

3
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1
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### Does a finite coresolution with Specht-filtered modules imply a Specht filtration?

$\newcommand{\sym}{\mathfrak{S}}$
$\newcommand{\rarr}{\rightarrow}$
Given a partition $\lambda$ of $n$ and a commutative ring $R$, writing $\sym_n$ for the symmetric group, there is a Specht module $S^...

4
votes

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295
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### Reference for fact about reduction mod $p$ of a representation of a finite group

Let $G$ be a finite group, let $M$ be a $\mathbb{Z}[G]$-module whose underlying abelian group is finite-rank free abelian, and let $\mathbb{F}$ be a field of characteristic $0$. For some prime $p$ ...

4
votes

1
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### Let A be an Artin algebra. What happens if the limit and inverse limit are the same in mod A?

Let $A$ be an Artin algebra and $\text{mod}\,A$ the category of finite length modules. Further, let $X_0 \longrightarrow X_1 \longrightarrow X_2 \longrightarrow ...$ and $Y_0 \longleftarrow Y_1 \...

2
votes

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### Reference for the action of the Mullineux involution on a partition with an added good node

Could you help me to locate a reference in the literature to the following fact: if you act by the Mullineux involution $M$ on a partition $Q$ which is a union of a regular partition $P$ with an added ...

9
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1
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### What is the largest subcategory $C$ of a module category over an Artin algebra, such that $C$ is Krull-Schmidt (and abelian)? Does $C$ exist?

Let $A$ be an Artin algebra, $\text{Mod}\,A$ the category of $A$-modules and $\text{mod}\,A$ the category of finitely generated $A$-modules. It is well-known that $\text{mod}\,A$ is a Krull-Schmidt ...

2
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### Is there a roadmap to learning representation theory of finite group over finite field?

I've been wanted to learn some basic theories of the (non-semisimple) representation of the finite group over a finite field.
I have been guessing that the materials might be contained in the books on ...

8
votes

2
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266
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### Is it possible for the reduction modulo $p$ of an non-commutative semisimple algebra to be commutative?

Suppose that $I, X_1, \ldots, X_{d-1}$ are $n \times n$ matrices with integer entries whose $\mathbb{Z}$-span is a subalgebra of $\mathrm{Mat}_n(\mathbb{Z})$. Suppose that, thought of as a subalgebra ...

2
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0
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### Upper bound on decomposition numbers for the symmetric group in a block of weight $w$

The $p$-blocks of the symmetric group $S_n$ are labelled by pairs $(\gamma, w)$ where $\gamma$ is a $p$-core partition, $w \in \mathbb{N}_0$ is the weight of the block, and $|\gamma| + p w = n$. ...