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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.

6
votes
1answer
241 views

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
1answer
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
0answers
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})=\...
1
vote
0answers
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
votes
1answer
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 ...
7
votes
0answers
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
votes
0answers
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
vote
0answers
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
votes
0answers
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 ...
1
vote
0answers
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
votes
1answer
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
1answer
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
votes
0answers
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
3answers
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 ...
1
vote
0answers
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
2answers
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
0answers
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
1answer
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
1answer
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 ...
6
votes
1answer
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
1answer
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 ...
6
votes
1answer
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
0answers
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
0answers
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
1answer
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\...
1
vote
0answers
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
1answer
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
1answer
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 ...
6
votes
1answer
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
1answer
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 ...
13
votes
1answer
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
3answers
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 $d$ ...
4
votes
0answers
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 ...
3
votes
1answer
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
3answers
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
1answer
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
1answer
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 ...
2
votes
0answers
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
1answer
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-...
4
votes
2answers
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 $G$...
2
votes
2answers
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 $\...
3
votes
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
1answer
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
3answers
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). ...