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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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 $...
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9 votes
2 answers
558 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 ...
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5 votes
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 $\operatorname{char} F$ ...
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3 answers
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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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3 votes
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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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10 votes
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230 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) ...
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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 ...
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8 votes
2 answers
374 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 \...
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9 votes
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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{...
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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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3 votes
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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 $k\sym_b$-...
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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^...
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4 votes
1 answer
180 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 $p$ ...
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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 \...
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2 votes
1 answer
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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 ...
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9 votes
1 answer
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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 ...
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0 votes
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164 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 ...
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8 votes
2 answers
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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 ...
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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 $|\gamma| + p w = n$. ...
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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 $\mathcal{H}(G)$ ...
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1 vote
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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 ...
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7 votes
1 answer
218 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 ...
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21 votes
1 answer
833 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 ...
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1 vote
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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 ...
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5 votes
0 answers
130 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 ...
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4 votes
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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 ...
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0 votes
0 answers
66 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 ...
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5 votes
1 answer
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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 ...
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2 votes
0 answers
105 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 ...
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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(\...
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  • 423
4 votes
1 answer
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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 $\sigma^{(m)}$ ...
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6 votes
1 answer
146 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 ...
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12 votes
0 answers
342 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 ...
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4 votes
1 answer
209 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 = \...
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3 votes
0 answers
89 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.
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  • 95
3 votes
1 answer
181 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-...
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0 votes
1 answer
116 views

Faithful representation of group of order $p^4$

In the (xi) group of the classification of groups of order $p^4$ given by W.Burnside in his book, "Theory of groups of finite order". The group ($\mathbb{Z}_{p^{2}}\rtimes \mathbb{Z}_{p^{}}) ...
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3 votes
1 answer
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Question concerning Brauer's second main theorem, Brauer correspondent blocks and blocks covered by nilpotent blocks

A version of Brauer's second main theorem is as follows: Let $G$ be a finite group, $x$ be a $p$-element of $G$, $B\in\mathcal{Bl}(G)$, and $\chi\in$ Irr$(B)$. If $d_{\chi\mu}^x\neq 0$ and $\mu$ ...
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12 votes
2 answers
512 views

Relations between quantum groups at roots of unity, modular representation theory, and physics

I understand that quantum groups at roots of unity are related to physics because they are used in the construction of Reshetikhin-Turaev invariants, conjectured by Witten. Are there other relations ...
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  • 2,189
2 votes
0 answers
80 views

Real non-principal 2-blocks for finite groups of Lie type

Is it true that each finite group of Lie type has a non-principal real $2$-block? Here the principal block is the one containing the trivial character and a block is real if it contains the complex ...
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3 votes
0 answers
81 views

Isomorphism of certain irreducible representations over finite fields

We are given a faithful representation of a cyclic group of order 5 $\rho: C_5=G \rightarrow End_{\mathbb{F}_3}(V) $ with $dim_{\mathbb{F}_3}V=8$ as vector space. It is also known that $V=U\oplus W$ ...
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  • 269
2 votes
1 answer
255 views

When is there a subring of the complex numbers surjecting onto a given field of prime characteristic?

To make use of the Lie algebra action of $\mathsf{gl}_2(\mathbb{C})$ to establish a isomorphism in modular representation theory, I would like an answer to this question: Let $K$ be a field of ...
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2 votes
1 answer
160 views

Are there always 1-dimensional projective representations

Let $G$ be a finite $p$-group and $(\mathbb{F},\mathcal{O},\mathbb{K})$ a sufficiently large $p$-modular system. Furthermore let $\mu_{p^\infty}\leq \mathcal{O}^\times$ the group of roots of unity of $...
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2 votes
1 answer
120 views

2-quotient of integer partition

This question is mostly about understanding the notation used in the following article: Alex Eskin, Andrei Okounkov, Pillowcases and quasimodular forms, in: Victor Ginzburg (ed.), Algebraic Geometry ...
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  • 621
3 votes
1 answer
140 views

Vanishing of first co-homology with coefficients modular representations of small dimension

Is the following true: For any $n$ there exists $p_0$ s.t. for any finite group $G$ of Lie type of rank $\leq n$ and characteristic $p\geq p_0$ and any (irreducible) $\mathbb F_p$ representation $V$ ...
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  • 2,185
11 votes
1 answer
437 views

conductor formula

Let $\pi_p$ be an irreducible representation of $GL_2(\mathbb{Q}_p)$. Assume $\pi_p$ is ramified,hence it will have a positive conductor. Consider $sym^3(\pi_p)$ which is a representation of $GL_4(\...
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  • 444
5 votes
0 answers
100 views

Extreme no loop conjecture for group algebras

Let $A=KG$ be a group algebra for a finite group $G$. Let $S$ be a simple $A$-module. The extreme no loop conjecture predicts that $Ext_A^1(S,S) \neq 0$ implies $Ext_A^i(S,S) \neq 0$ for infinitely ...
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  • 21.8k
1 vote
1 answer
69 views

Selfextensions for modules of commutative Frobenius algebras

Let $A$ be a commutative finite dimensional Frobenius algebra and $M$ a non-projective $A$-module. Can we have $Ext_A^i(M,M)=0$ for some $i>0$? Can we have $Ext_A^i(M,M)=0$ for some $i>0$ in ...
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  • 21.8k
12 votes
0 answers
491 views

Does $\mathrm{Ext}^1(M,M) \neq 0$ imply $\mathrm{Ext}^2(M,M) \neq 0$?

$\DeclareMathOperator{\Ext}{\operatorname{Ext}}$The first question is about group algebras: Question 1: Let $A=kG$ be a group algebra (with $G$ finite) and let $M$ be an indecomposable $A$-module. ...
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