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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Which finite group schemes over positive characteristic have projectivity detected by a set of (generalized) cyclic subgroups?
A finite algebra $B$ over a field $K$ has projectivity detected by a set $S = \{f_i : A \to B \mid i \in I\}$ of maps into $B$ from a fixed algebra $A$ if for any finite module $M$, $M$ is projective ...
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Question about defect subgroups
Let $G$ be a finite group over an algebraically closed field $\Bbbk$ of characteristic $p>0$. Let $b$ be a block of $G$. Then the a defect subgroup $D$ of $b$ is one for which every module in $b$ ...
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How does one compute the group action of the automorphism group on integral cohomology?
Suppose I have a curve $X$ (for concreteness, we can take $X$ to be a smooth, projective curve over a finite field $\mathbb F_q$, and even more concretely consider the family of curves described by ...
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Is there a way to find the eigenvalues of a matrix using character table?
I am studying applications of representation theory. I want to know if there is a procedure to find the eigenvalues and eigenvectors of a matrix using the character table of the Group acting on its ...
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What goes wrong with the Brauer construction for a module over a complete DVR?
Let $G$ be a finite group, $\mathcal{O}$ a complete discrete valuation ring, and $F$ the residue field. The Brauer construction for a fintely-generated $\mathcal{O}G$-module $M$ with respect to a $p$-...
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Exact structures on representations of a finite group
For simplicity assume $G$ is a (finite) $p$-group, and $k$ is field of characteristic $p$, so that there exists a unique simple $kG$-module the trivial module $k$. I am looking for a class of short ...
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Congruences between Eisenstein series and cusp forms
Let $k\geq 4$ be an even integer. Let $p>k$ be a prime
such that $p\mid B_k$, the $k$th Bernoulli number. Then there is a primitive cusp form $f=\sum_{n\geq1}c(n, f)q^n$
of weight $k$ and level $1$ ...
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Do you know a survey of modular Lie algebras and its representations?
When I was an university student, I liked reading some books about the representation theory of finite groups or Lie algebras and I was interested in explicit constructions of irreducible ...
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Fixed points of a linear abelian p-group in characteristic p
Let $V=\bigoplus_{n\geq1}\mathbb F_{p}\cdot e_{n}$ be an $\mathbb F_{p}$-vector space of countable dimension, and write $V_{n}=\operatorname{Vect}(e_{1},\dotsc,e_{n})$. Let $G$ be a (possibly infinite)...
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Using the mapping cone to show that a chain map defines a stable equivalence between two symmetric algebras
This question is about an argument in the proof of Theorem 9.8.8 in Linckelmanns Block Theory of Finite Group Algebras. I need to understand the argument in order to do something similar in my ...
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Can modular representation theory be used to prove Sylow's existence theorem?
Edit 20/12: I added a more precise question at the bottom of the post.
Given a finite group $G$ and a prime $p$, we want to prove that $G$ has a $p$-subgroup $P$ such that $|G:P|$ is not divisible by $...
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The mysterious significance of local subgroups in finite group theory
EDIT 21/12: Even if there are no conclusive answers to these questions, I would very much like to know if anyone has noted and attempted to explain the mysterious significance of local subgroups: are ...
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Classification of restricted Lie algebras of reductive groups
$\DeclareMathOperator\Lie{Lie}$Let $G/K$ be a reductive group over a field $K$. In characteristic $0$ the Lie algebra is invariant under base change of fields, so to understand $\Lie(G)$ it is enough ...
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What is $\dim D^{\lambda}$ for the symmetric group?
What are the dimensions of the simple modules $D^{\lambda}=S^{\lambda}/S^{\lambda}\cap (S^{\lambda})^{\perp}$ for the modular representation theory of $S_n$, i.e. $\operatorname{char}(k)=p>0$?
I ...
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Asymptotics for number of $p$-regular partitions of $n$
The number of simple modules $D^{\lambda}=S^{\lambda}/S^{\lambda}\cap (S^{\lambda})^{\bot}$ of the symmetric group over a field $k$ such that $\text{char}(k)=p > 0$ is the number of $p$-regular ...
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