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

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6
votes
3answers
274 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
76 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\ ...
1
vote
1answer
155 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 ...
1
vote
0answers
104 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 = ...
6
votes
1answer
276 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 ...
4
votes
2answers
550 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 ...
0
votes
0answers
70 views

classical specializations of Hida families

Let 𝕋 denote the ordinary Λ-adic Hecke algebra of say tame level N. If $P_{F}$ minimal prime of 𝕋 correspond to hida families F and this family specialise to classical weight one ordinary modular ...
2
votes
2answers
148 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
226 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; ...
4
votes
1answer
126 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
253 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
137 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 ...
1
vote
1answer
228 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
229 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
295 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
198 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
383 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). ...
1
vote
1answer
257 views

Defect groups and subgroups

I have asked this question on Stack Exchange but had no response; it's been bugging me for a few days. I am struggling to see how to apply Mackey's theorem to prove a certain Lemma in Local ...
4
votes
1answer
389 views

Can the projection (tensor algebra) -> (symmetric algebra) be forced to split in char. p by factoring out p-th powers?

Question 1 (the weak and simple statement, which, I think, already is wrong): Let $p$ be a prime. Let $k$ be a field with characteristic $p$. For any $k$-vector space $V$, consider the canonical ...
2
votes
2answers
820 views

Projective modules and tensor products

My question(s) relate(s) to pp51-52 of Local Representation Theory by JL Alperin -- the relevant pages are contained in the Google Books preview ...
1
vote
0answers
299 views

Do all finite $W$-superalgebras have 1-dimensional representations?

Premet proved the famous KW-conjecture in modular Lie algebra. After, Premet introduced the finite $W$-algebra $U(g, e)$. Also, Premet proposed the conjecture every algebra $U(g, e)$ admits a ...
4
votes
3answers
381 views

Molien for modular representations?

Let $G$ be a finite group, and let $k$ be a field whose characteristic divides $\left|G\right|$. Let $\rho:G\to \mathrm{End} V$ be a (finite-dimensional) representation of $G$ over $k$. Prove or ...
3
votes
3answers
691 views

Innocent question on tensor products of modular representations

Let $K$ be a field (of course, of positive characteristic, unless you want a trivial question). Let $G$ be a finite group, and $V$ and $W$ be two completely reducible (finite-dimensional) ...
4
votes
2answers
684 views

two questions in modular representation theory

I have two questions: Let $G$ be a finite group. Because complex representations of $G$ are completely reducible to know all representations is same as knowing irreducible ones. In case of modules ...
6
votes
3answers
573 views

Exact sequences of permutational representations?

Let $R$ be a commutative ring, like the ring of integers $\mathbb Z$ or the ring of $p$-adic integers $\mathbb Z_p$. Let $G$ be a finite group; let us consider permutational representations of $G$ ...
7
votes
3answers
694 views

Compact generation for modular representations

Are the derived categories of modular representations of algebraic groups compactly generated? (e.g. consider SL_2 in characteristic 2). Note modular reps of finite groups are compactly generated (by ...