All Questions
17 questions
14
votes
0
answers
298
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) ...
- 1,389
9
votes
1
answer
355
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 ...
- 239
6
votes
3
answers
505
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). ...
- 1,269
6
votes
1
answer
186
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 ...
- 26.5k
6
votes
1
answer
366
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 $r$-...
- 367
5
votes
2
answers
347
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 ...
- 53
5
votes
0
answers
149
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 ...
- 26.5k
4
votes
1
answer
321
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 $p$ ...
- 41
4
votes
0
answers
107
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 ...
- 26.5k
3
votes
1
answer
95
views
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 $\mu$ ...
- 1,755
3
votes
0
answers
91
views
Are the reductions of the cuspidal characters of GL2(Fq) distinct?
Let $p$ be an odd prime and $q=p^n$ for some $n \geq 1$. If $\mathbb{F}_q$ is the unique, up to isomorphism, finite field with $q$ elements then the cuspidal representations of the group $\rm{GL}_2(\...
- 117
3
votes
0
answers
104
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.
- 145
2
votes
1
answer
75
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 ...
2
votes
0
answers
382
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 ...
- 1,013
1
vote
0
answers
83
views
$p$-modular splitting systems and the characteristic of the ring $\mathcal{O}$
Let $k=\overline{k}$ be a field of characteristic $p$.
Let $(K,\mathcal{O},k)$ be a $p$-modular system.
Let both $k$ and $K$ be splitting fields for $G$ and its subgroups.
The ring $\mathcal{O}$ can ...
- 311
0
votes
1
answer
143
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^{}}) ...
- 381
0
votes
0
answers
289
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 ...
- 2,179