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Questions tagged [modular-representation-theory]

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

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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 $b$ ...
freeRmodule's user avatar
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4 votes
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111 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 ...
Asvin's user avatar
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114 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 ...
starry's user avatar
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5 votes
1 answer
367 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 $p$-...
jb2g4's user avatar
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3 votes
0 answers
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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 ...
Justin Bloom's user avatar
3 votes
0 answers
160 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 $1$ ...
Zakariae.B's user avatar
7 votes
1 answer
311 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 ...
Ozzie's user avatar
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5 votes
2 answers
466 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)...
abeaumont's user avatar
2 votes
1 answer
150 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 ...
jb2g4's user avatar
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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 $...
semisimpleton's user avatar
17 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 ...
semisimpleton's user avatar
2 votes
0 answers
122 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 ...
Martin Ortiz's user avatar
4 votes
0 answers
284 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 ...
Jackson Walters's user avatar
3 votes
1 answer
211 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 ...
Jackson Walters's user avatar
5 votes
0 answers
240 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(\...
Jackson Walters's user avatar
5 votes
1 answer
231 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....
Sunny Sood's user avatar
2 votes
0 answers
162 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 ...
user avatar
1 vote
1 answer
186 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 ...
user avatar
1 vote
0 answers
79 views

$p$-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 ...
Stein Chen's user avatar
24 votes
0 answers
783 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 ...
semisimpleton's user avatar
3 votes
0 answers
145 views

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{\...
An Zhang's user avatar
2 votes
0 answers
61 views

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 ...
skupers's user avatar
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2 votes
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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, ...
JBuck's user avatar
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6 votes
1 answer
140 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 ...
Mare's user avatar
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1 vote
0 answers
86 views

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 ...
kindasorta's user avatar
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2 votes
0 answers
129 views

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 ...
IntegrableSystemsEnthusiast's user avatar
2 votes
0 answers
45 views

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 ...
IntegrableSystemsEnthusiast's user avatar
3 votes
0 answers
59 views

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 ...
IntegrableSystemsEnthusiast's user avatar
9 votes
0 answers
297 views

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 ...
Kimyeong Lee's user avatar
4 votes
0 answers
161 views

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 ...
Mattia's user avatar
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2 votes
0 answers
52 views

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 $...
John McHugh's user avatar
9 votes
2 answers
730 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 ...
Doron Grossman-Naples's user avatar
5 votes
1 answer
233 views

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 $\operatorname{char} F$ ...
Uriya First's user avatar
  • 2,846
6 votes
3 answers
1k 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 ...
DaveWasHere's user avatar
3 votes
0 answers
218 views

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 ...
death_cube_k's user avatar
5 votes
0 answers
106 views

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 ...
Ehud Meir's user avatar
  • 5,009
13 votes
0 answers
274 views

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) ...
Arthur's user avatar
  • 1,379
11 votes
2 answers
835 views

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 ...
WJL's user avatar
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8 votes
2 answers
431 views

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 \...
N. SNANOU's user avatar
  • 383
9 votes
0 answers
431 views

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{...
Victor TC's user avatar
  • 795
4 votes
0 answers
170 views

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&...
kevkev1695's user avatar
  • 1,023
3 votes
0 answers
133 views

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 $k\sym_b$-...
Cihan's user avatar
  • 1,606
3 votes
1 answer
97 views

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^...
Cihan's user avatar
  • 1,606
4 votes
1 answer
295 views

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 $p$ ...
Gina's user avatar
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4 votes
1 answer
201 views

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 \...
kevkev1695's user avatar
  • 1,023
2 votes
1 answer
72 views

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 ...
IntegrableSystemsEnthusiast's user avatar
9 votes
1 answer
208 views

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 ...
kevkev1695's user avatar
  • 1,023
2 votes
0 answers
349 views

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 ...
gualterio's user avatar
  • 1,053
8 votes
2 answers
266 views

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 ...
Mark Wildon's user avatar
  • 10.8k
2 votes
0 answers
78 views

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 $|\gamma| + p w = n$. ...
Mark Wildon's user avatar
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