# 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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### first group cohomology for the standard representation of $S_n$ over $\mathbb{F}_2$

Let $g \geq 2$ be an integer and consider the symmetric group $S_n$ where $n = 2g+1$ or $n = 2g+2$ as a subgroup of the symplectic group $\mathrm{Sp}_{2g}(\mathbb{F}_2)$ via the standard ...

**6**

votes

**1**answer

175 views

### Why is Nagao's theorem the “Module theoretic version of Brauer's second main theorem”?

Let $G$ be a finite group, $p\in\mathbb{P}$ a prime, $\mathbb{F}$ an algebraically closed field of characteristic $p$, and $D\leq G$ a $p$-subgroup.
Brauer second main theorem states
If $\chi\in ...

**1**

vote

**0**answers

43 views

### To find $\dim(D^{\mu})=\dim\frac{S^{\mu}}{S^{\mu}\cap S^{\mu\perp}}$

Let $S_6$ be the symmetric group of degree $6$ and $F$ be any finite field of characteristic $2.$ Then $2$-regular partition of $6$ are $(5,1)$, $(4,2)$ and $(3,2,1)$ . I have to find $$\dim(D^{\mu})=\...

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21 views

### Decomposition numbers of characteristic 0 Ariki-Koike algebras for q=0 or q generic

Consider a complex Ariki-Koike algebra $H:=H(q;Q_1,\dots, Q_\ell)$ for some invertible elements $q, Q_1,
\dots, Q_\ell$ of $\mathbb{C}$, i.e. an associative $\mathbb{C}$-algebra with generators $T_0, \...

**5**

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**1**answer

162 views

### Questions on group and Nakayama algebras from a book

Recall that a Nakayama algebra (also called serial algebras in the literature) is an algebra such that every indecomposable module has a unique composition series.
In the book "Classical artinian ...

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**0**answers

101 views

### Modular version of Mednykh's formula?

Let $G$ be a finite group and $\Sigma_g$ a closed Riemann surface of genus $g$. Then Mednykh's formula states
$\frac{\left|\mathrm{Hom}(\pi_1(\Sigma_g),G)\right|}{\left|G\right|} = \frac{1}{\left|G\...

**2**

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36 views

### Branching rule for degenerate cyclotomic Hecke algebras

Grojnowski and Vazirani showed that there is a crystal isomorphism between the crystal defined by modular branching of simple modules of Ariki-Koike algebras and that of an integral highest weight ...

**1**

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36 views

### Why are the definitions of i-good nodes of a multipartition equivalent?

Let $e\geq 2$ and $0\leq i\leq e-1$.
For a multipartition $\lambda\vdash_\ell n$ of $n $ one can define the notion of an $i$-good box of the Young diagram of $\lambda$. But there are seemingly ...

**3**

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116 views

### Questions about ``$p$-canonical basis" for $\widehat{\mathfrak{sl}_n}$ module (wedge power of natural representation)

Let $p$ be a prime number. Consider the natural representation of the affine Lie algebra $\widehat{\mathfrak{sl}_p}$, defined as follows.
$$A = \bigoplus_{i=1}^N \mathbb{C}a_i; \qquad \text{nat}_p = A ...

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87 views

### $G$-invariant bilinear maps

Let $G$ be a finite group and $M$ a finitely generated $\mathbb{Z}G$-module. Then $M \otimes k$ is a representation of $G$ for any field $k$. I am interested in the number of ways we can turn $M \...

**11**

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**1**answer

305 views

### Can we glue characteristic 0 and characteristic p representations of a finite group given equality of (Brauer) characters?

Suppose I have a prime $p$ and a finite group $G$ together with representations $\sigma: G \to GL_n(\mathbb{Q}_p)$ and $\pi: G \to GL_n(\mathbb{F}_p)$. My question is:
When does there exist a ...

**4**

votes

**1**answer

72 views

### Sufficient conditions for secondary invariants

Let $G$ be a finite group, $k$ be a field whose characteristic divides $|G|$, and $\rho:G\hookrightarrow\operatorname{GL}_n(k)$ be a faithful representation of $G$. Let $V$ be a $k$-space of dimension ...

**4**

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114 views

### subgroups of $\mathrm{Sp}_{2g}(\mathbb{Z}_2)$ whose mod-2 image is the symmetric group

Let $G \subseteq \mathrm{Sp}_{2g}(\mathbb{Z}_2)$ be a closed subgroup of the symplectic group over the $2$-adic integers whose image under the mod-$2$ homomorphism $\pi : \mathrm{Sp}_{2g}(\mathbb{Z}_2)...

**10**

votes

**3**answers

414 views

### Low-dimensional irreducible 2-modular representations of the symmetric group

I apologize if this question is a little too basic for MathOverflow, but it's somewhat outside of my background and I'm frustrated that the answer doesn't seem to be explicit in the literature even ...

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vote

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41 views

### Largest almost quasisimple group that acts on a spin module

I'd like to know the largest almost quasisimple group $G$ which acts faithfully and irreducibly on the spin module for an odd dimensional orthogonal group $\Omega(2k+1,q)$, or on each of the two half-...

**5**

votes

**2**answers

311 views

### Module with indecomposable and decomposable reductions mod $p$

Let $G$ be a finite group and $p$ a prime dividing the order of $G$. Let $V$ be an irreducible $\mathbb{C}[G]$-module.
Let $F$ be the finite field $\mathbb{Z} / p \mathbb{Z}$. Suppose that there ...

**13**

votes

**0**answers

200 views

### Group homology $\mathrm{SL}_2$ acting on $\mathrm{Sym}^g$

Let $k$ be a field. We write $\mathrm{Sym}^g(k^2)$ for the $g$-th symmetric power of the (a?) standard representation of $\mathrm{GL}_2(k)$ ($g\geq 0$ an integer). Here I consider $\mathrm{Sym}^g(k^2)$...

**2**

votes

**1**answer

118 views

### How to prove that $M^G=\mathbb{F}_p[x_1\cdot v, x_1^{p\cdot(p-1)}+ v^{p-1}]$?

Let $G=Sl_2(\mathbb{F}_p)$ and $M= \mathbb{F}_p[x_1,x_2]$, where $p$ is a prime.
$M$ is a $G$-module with $(A\cdot x_1, A\cdot x_2)=(x_1,x_2)\cdot A, (\forall) A \in Sl_2(\mathbb{F}_p)$.
I have to ...

**2**

votes

**1**answer

221 views

### Why we study Endo-Trivial Modules?

I made the exact same question on MSE some days ago and there wasn't any response whatsoever so far, so thought probably I have to ask here to get an answer. So the question goes as follows:
Recently ...

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votes

**1**answer

302 views

### Group of order $5p^aq^b$

In Lectures by Dan Bump on Modular representation theory,
Theorem 13.14 states that whenever $G$ is a non-abelian simple group of order $|G|=p^aq^br$ for distinct primes $p$,$q$, and $r$, every $r$-...

**0**

votes

**1**answer

55 views

### Representations of smash products with $p$-groups

I am trying to find more generalized counterparts of some well-known results from modular group representations.
My question is the following:
Suppose that $H$ is a finite $p$-group acting as ...

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votes

**1**answer

375 views

### Finite groups with all irreducible representations one dimensional

Let $k$ be a field of characteristic $p$ and $G$ a finite group. This question might be a dulicate of this question:
Which finite groups have no irreducible representations other than characters?
...

**0**

votes

**0**answers

121 views

### Group representations over fields of non-zero characteristic

I try to read some aspects of modular representation theory and I have some questions I have encountered, which are more concerned with some basic number theory, I guess.
I understand that the ...

**1**

vote

**0**answers

63 views

### Why is the kernel of the Brauer homomorphism a module for the normaliser?

Let $G$ be a finite group and $k$ a field. The Brauer morphism is defined as the map
$$M\mapsto Br_P(M):=\frac{M^P}{\sum_Q \text{tr}^P_Q(M^Q)},$$
where $Q<P$ where $M^P$ is the points in $M$ fixed ...

**4**

votes

**1**answer

152 views

### Projectives in the category of modular representations of Lie algebras

Let $\mathfrak{g}$ be a semi-simple Lie algebra (eg. $\mathfrak{g} = \mathfrak{sl}_n$), defined over an algebraically closed field $\textbf{k}$ with characteristic $p >> 0$. The center $Z(U\...

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vote

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84 views

### derived invariants, perversity and modular coefficients

Let $\pi:X\rightarrow Y$ a Galois cover of finite type schemes over $\mathbb{C}$ of group $\Gamma$.
Let $n$ an integer such that it is not prime with the order of $\Gamma$.
Then $\pi_{*}\mathbb{Z}/n\...

**6**

votes

**1**answer

194 views

### Real-valued character in Block with cyclic defect has at most two constituents modulo $p$

Let $G$ be a finite group and let $(K,R,k)$ be a $p$-modular system (large enough for $G$ etc.) and consider a block algebra $B \subseteq RG$ with cyclic defect group.
My question is about the ...

**2**

votes

**1**answer

159 views

### indecomposable modules restricted from $gl_n$ to $sl_n$

Let K be an algebraic closed field, $gl_n$ be the general linear Lie algebra over K, and $sl_n$ be the special linear Lie algebra.
Let $\chi\in gl_n^*$. Let $U_\chi(gl_n)$ be the corresponding reduced ...

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votes

**1**answer

303 views

### Well-understood bases for Grothendieck groups of modular representation categories

Let $\mathfrak{g}$ be a semi-simple Lie algebra.
So in characteristic $0$, the Grothendieck group of a block of category $\mathcal{O}$ is given by the classes of the Verma modules. Unlike the ...

**8**

votes

**1**answer

303 views

### Jordan-Hölder-like statements for modules with $\Delta$-filtrations over a quasihereditary algebra

Definitions
Let $A$ be an Artin algebra (for instance, take $A$ to be a finite dimensional algebra over some field) and label the isomorphism classes of simple $A$-modules by the elements of a ...

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votes

**1**answer

1k views

### Which finite groups have no irreducible representations other than characters?

A classical result states that all the irreducible representations of a finite group over $\mathbb{C}$ are characters if and only if $G$ is abelian. I would like to know what happens if we consider a ...

**14**

votes

**3**answers

1k views

### A question on representation of graphs

Take a complete graph $K_n$. You want to assign a vectors from $\Bbb F_2^d$ to every edge such that sum of vectors in every simple cycle does not sum to $0$ vector. The question is what is minimum $d$ ...

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votes

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236 views

### Decomposition of symmetric powers of reduced regular representation modulo $p$

Let $\bar{\rho}$ denote the reduced regular representation of $\mathbb{Z}/p$ over a field of characteristic $p$. The representation $\mathrm{Sym}^k \bar{\rho}$ decomposes (for each $k$) as a sum of ...

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**1**answer

221 views

### the number of indecomposable modules of finite groups over finite fields of a fixed dimension

I am interested in determining the the number of indecomposable modules of finite groups over finite fields of a fixed dimension. Specifically, I have the following conjecture:
Conjecture. Suppose we ...

**6**

votes

**3**answers

373 views

### Restriction of scalars for the adjoint representation of $SL_2(\mathbb F_q)$

Let $p$ be a prime number, $q=p^e$ a power of $p$, and $G=SL_2(\mathbb F_q)$. Let $V$ be the adjoint representation of $G$, i.e. $V$ is the 3-dimensional $\mathbb F_q$-space of of (2,2)-matrices of ...

**3**

votes

**1**answer

100 views

### Idempotents and Structure of Simple GL(n,p) modules in the describing characteristic

Let $F_2$ denote the finite field of two elements, and $GL(n,2)$ the general linear group of degree $n$ over $F_2$. I have a promising line of inquiry into identifying the structure of simple $F_2\ GL(...

**2**

votes

**1**answer

390 views

### Does Schur's Lemma hold in this case? Regular representations of $S_n$ over $\mathbb R$

Warning I am a physicist and I am not familiar with a lot of the machinary of representation theory.
I consider the regular representation of $\mathbb S_n$ over reals $\mathbb R$ ($\mathbb R \mathbb ...

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votes

**0**answers

299 views

### Actions and representations of profinite groups

Let $p$ be a prime number, and denote by $\mathbb{Z}_p$ the additive profinite group of p-adic integers. Let $G$ be a finitely generated profinite group of order coprime to $p$, and $V = \mathbb{Z}_p^{...

**6**

votes

**1**answer

631 views

### What is the Grothendieck group of the category of $\mathbf{Z}_p[G]$-modules?

Let $G$ be a finite group. Let $\mathcal{O}$ be a suitably large finite extension of the $p$-adic integers, with residue field $\mathbf{F}_q$.
The Grothendieck group of the category of finitely-...

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votes

**2**answers

800 views

### Decomposing representations of finite groups

Let $G$ be a finite group, $p$ a prime number. We denote by $\mathbb{F}_p$ the field of cardinality $p$. Let $V$ be an infinite dimensional representation of $G$ over $\mathbb{F}_p$.
Must there be $G$...

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votes

**2**answers

247 views

### Composition factors of tensor products of modular representations

In ordinary representation theory over $\mathbb{C}$, all the irreducible modules of a finite group $G$ appear as composition factors of the tensor products $X \otimes \cdots \otimes X$ of a faithful $\...

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**0**answers

280 views

### Rejects and injectives

Let $A$ be any ring and consider modules on the left. For $M$ $A$-module, the trace $Tr(M,A)$ is a two-sided ideal of $A$. If $A$ is a unitary ring then:
$Tr(P,A)P=P$, for $P$ projective;
$Tr(P,A)^2=...

**4**

votes

**1**answer

144 views

### Categorified versions of Mackey's functor

I would like to ask for possible references for the following very general situation, a categorified version of Mackey functors.
The question is if there are other known constructions to associate to ...

**3**

votes

**1**answer

352 views

### Brauer homomorphism and simple modules

Hey there,
several weeks ago, there was a discussion on the Brauer hom (see Is the Brauer correspondence injective ? ). I like to investigate this hom when being applied to simple modules:
Let $k$ ...

**0**

votes

**1**answer

159 views

### Analogon to Brauer characters, if K not algebraically closed

Hello,
is there a theory for characters of a finite group over a field $K$ with prime characteristic, if $K$ is not algebraically closed? For algebraically closed fields $K$ for example Brauer found ...

**2**

votes

**1**answer

343 views

### Representations of semidirect product over $C_p$

Hi,
I am wondering if anything is known about irreducible representations of a semidirect product over $C_p:=\mathbb{Z} / p \mathbb{Z}$ in general or at least in special cases. For example of $C_q \...

**0**

votes

**0**answers

261 views

### Modular representations of the symplectic group

Let G=Sp(2m,2) be a finite symplectic group acting on $F_2^{2m}$. This group G acts 2-transitively on $\Omega_{+}$ and on $\Omega_{-}$. Let $F$ be an algebraic closure of $F_2$.
I am interested to ...

**4**

votes

**1**answer

373 views

### What do we know about periodic modules in p-groups?

Hi,
a module in KG,where G is a p-group and K a field of characteristic p, is called periodic if $ \Omega^{n} M = M $, for a natural n.
In general the full subcategory of periodic modules seems to ...

**6**

votes

**1**answer

242 views

### What are the irreducible modular representations of $SU(n,p)$?

To fix notation, by $SU(n,p)$, I mean the subgroup of $SL_n(\mathbb F_{p^2})$ consisting of matrices $A$ which satisfy $\overline A^t A = 1$, where $\overline A$ is the matrix given by raising all the ...

**6**

votes

**3**answers

445 views

### Irreducible mod-p representation of a semidirect product with trivial p-core

Consider the group $G=H\rtimes{}C$, where $H$ has order prime with $p$ and $C$ is cyclic of order $p^k$, and $C\rightarrow{}\mathrm{Aut}(H)$ is faithful (or equivalently $G$ has trivial $p$-core). ...