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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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
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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, ...
JBuck's user avatar
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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 ...
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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 ...
kindasorta's user avatar
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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 ...
IntegrableSystemsEnthusiast's user avatar
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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 ...
IntegrableSystemsEnthusiast's user avatar
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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 ...
IntegrableSystemsEnthusiast's user avatar
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157 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
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Irreducibility of adjoint representation

Let $ \mathbb{F} $ be a finite field of characteristic $ p\geq 5 $, $ G $ a finite group and $ \rho:G\to {\rm GL}_{2}(\mathbb{F}) $ be a representation of $ G $. By $ \text{ad}^{0}(\rho) $ we denote ...
Nobody's user avatar
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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 ...
Mattia's user avatar
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Modularity and Galois representations notation

Reading some literature concerning modularity of elliptic curves, more particularly the study of the corresponding Galois representations I sometimes see $$\rho_{E,p} \simeq \rho_{\mathfrak{f},p}$$ ...
did's user avatar
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$p$-centric subgroups control fusion

Given two finite groups $G,H$ such that $$\displaystyle\bigoplus_{Q\cong L\leq G\ \mathrm{up\ to}\ G\mathrm{-conjugation}}{\mathbb{F}_p\Bigg[\frac{\mathrm{Out}(L)}{\mathrm{Out}_G(L)}\Bigg]}\cong\...
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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 $...
John McHugh's user avatar
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637 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
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1 answer
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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 $\operatorname{char} F$ ...
Uriya First's user avatar
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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 ...
DaveWasHere's user avatar
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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 ...
death_cube_k's user avatar
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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) ...
Arthur's user avatar
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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 ...
WJL's user avatar
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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 \...
SAID's user avatar
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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{...
Victor TC's user avatar
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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&...
kevkev1695's user avatar
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101 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
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3 votes
1 answer
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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^...
Cihan's user avatar
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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 $p$ ...
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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 \...
kevkev1695's user avatar
2 votes
1 answer
71 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
189 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 vote
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225 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
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8 votes
2 answers
249 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
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2 votes
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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 $|\gamma| + p w = n$. ...
Mark Wildon's user avatar
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3 votes
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compactly induction of smooth modules over Hecke algebras

Let $G$ be a locally profinite group and $H$ its closed subgroup. It is well-known that a smooth representation of $G$ is identified with a smooth module over $\mathcal{H}(G)$, where $\mathcal{H}(G)$ ...
M masa's user avatar
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1 answer
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Irreducible Representation of A_5

Knowing the fact that standard representation arising out of permutation representation of $A_5$ over $\mathbb{C}$ is irreducible and of degree $4$. What can we conclude about the irreducibility over ...
Himanshu Setia's user avatar
8 votes
1 answer
265 views

Reference request for indecomposable representations of $\mathfrak{sl}(2)$ over an algebraically closed field of characteristic $p > 0$

Is there a good reference on the classification of indecomposable representations of the Lie algebra $\mathfrak{sl}(2)$ over an algebraically closed field of characteristic $p > 0$ (of course, in ...
tyrese's user avatar
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22 votes
1 answer
975 views

How to see that the determinant of this matrix is nonzero for all primes?

I'm trying to show that $\sum_{i = 0}^{p-2} (i+1)^{-1} t^{i+n}$ where $0 \leq n \leq p-2$ spans the vector space $\mathbb{F}_p[t]/(1-t)^{p-1}$ as a rank $p-1$ module over $\mathbb{F}_p$. In other ...
Catherine Ray's user avatar
1 vote
0 answers
51 views

When is a block of a group algebra a trivial extension algebra?

Let $B$ be a block of a group algebra. See for example https://math.stackexchange.com/questions/229412/trivial-extension-of-an-algebra for the definition of a trivial extension algebra (which is ...
Mare's user avatar
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5 votes
0 answers
140 views

Quiver and relations of $F\mathrm{SL}(2,q)$

$\DeclareMathOperator\SL{SL}$Let $q=p^n$ be a prime power and $F$ a field of characteristic two. Let $G=SL(2,q)$ the group of $2 \times 2$ special linear matrices over the field with $q$ elements with ...
Mare's user avatar
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4 votes
0 answers
101 views

Which algebras of quaternion type do appear as blocks of group algebras?

A finite dimensional algebra $A$ is said to be of quaternion type (first defined by Erdmann?) when it is tame, symmetric and connected and furthermore the Cartan matrix is nonsingular and the stable ...
Mare's user avatar
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0 votes
0 answers
90 views

Decomposition an $A$-module to irreducible ones

Let $V$ be a complex vector space (i.e, over the filed of complex numbers) and $A$ be a complex algebra. Suppose that $V$ is an $A$-module. Under what proper condition(s) there are irreducible ...
ABB's user avatar
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5 votes
1 answer
205 views

Representations of $\operatorname{Sp}(2g,\mathbb{Z}_3)$

Let $V$ be a $2g$-dimensional vector space over $\mathbb{Z}_3 := \mathbb{Z}/3\mathbb{Z}$. First, $\operatorname{Sp}(2g,\mathbb{Z}_3)$ acts on $\Lambda^2(V)$, and this decomposition is reducible, as ...
Quentin Faes's user avatar
2 votes
0 answers
110 views

Loewy structure of $S_4$

How to deduce the Loewy Structure of $kS_4$ where $k$ has characteristic 2. I can compute the Cartan matrix and Decomposition matrix with Brauer Characters without difficulties. But when it comes to ...
user666's user avatar
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4 votes
0 answers
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No irreducible subrepresentations

Let $p$ be a prime number. In the paper The image of Colmez's Montreal functor, on the page 12, the author points out an interesting property: The smooth mod $p$ representation $\mathrm{c-Ind}^{GL_2(\...
A413's user avatar
  • 423
4 votes
1 answer
88 views

Field of definition for isomorphism classes of modular representations

Let $G$ be a finite group, and let $\sigma: G \to GL_n(k)$ be a (not necessarily irreducible) representation defined over an algebraically closed field $k$ of characteristic $p$. Let $\sigma^{(m)}$ ...
Nate's user avatar
  • 1,859
6 votes
1 answer
166 views

Endomorphism ring of trivial source modules for abelian p-groups

Bernhard Böhmler  (who is also on MO) and myself had the following idea: Let $G$ be a finite group and $k$ a field of characteristic $p$ (algebraically closed when it is needed) such that $p$ divides ...
Mare's user avatar
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12 votes
0 answers
385 views

Non-isomorphic groups with same character tables and different Brauer character tables

Let $A$ be a discrete valuation ring with perfect residue field $k$ of characteristic $p$ and field of fractions $K$ of characteristic $0$. Let $G$ and $H$ be two finite groups and assume that $K$ is ...
Sebastian Spindler's user avatar
4 votes
1 answer
216 views

Does any group action of $\mathbb{F}_p^\mathbb{N}$ on $\mathbb{F}_p^\mathbb{N}$ have non-trivial fixed points?

It is well-known that if $G$ is a finite $p$-group acting on a non-zero $\mathbb{F}_p$-vector space $V$, then $V^G \neq \{0\}$. My question is about a generalization of this result when $G = V = \...
Arkandias's user avatar
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3 votes
0 answers
91 views

Reference for representations of $\text{GL}_n(\mathbb{Z}/p^k\mathbb{Z})$ over $\mathbb{F}_p$

I was looking for a reference on representations of $GL_n(\mathbb{Z}/p^k\mathbb{Z})$ over $\mathbb{F}_p$. In particular what the irreducible and indecomposable representations look like.
Niareh's user avatar
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3 votes
1 answer
212 views

Fourier transform on finite groups in characteristic $p>0$

Is there a Fourier theory for finite groups in characteristic $p>0$? Assume that $p$ divides the order $|G|$ of finite groups (or just work with $p$-groups), i.e., in a modular representation-...
l'etranger's user avatar