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

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Background: Let $G$ be a finite group. Fix a prime $p$. We say that $H\subseteq G$ is strongly $p$-embedded if: $p$ divides $|H|$; For every $x\in G-H$, $p$ does not divide $|H\cap H^x|$. Some facts ...
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Let $G$ be a finite group, and fix a prime $p$ which divides $|G|$. The ordinary (complex) characters $\text{Irr}(G)$ can be partitioned into what are called $p$-blocks, and to each block $B$ can be ...
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Let $k$ be a field of characteristic $p>0$. Let $G$ be a finite group with $p\mid |G|$. For each conjugacy class $K\in\text{cl}(G)$, let $x_K\in K$ be a representative, let $C_K=\mathbf{C}_G(x_K)$, ...
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My ongoing progress is about representation theory and number theory, to be more specific, modular representation of General linear groups over local field. My advisor ask me to submit a reading-list ...
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Let $(F,R,k)$ be a splitting $p$-modular system for a finite group $G$. (Here, $R$ is a discrete valuation ring with residue field $k$ of characteristic $p$ and field of fractions $F$.) Let $U$ be an $...
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Let $G$ be a finite group, $k$ a characteristic $p$ field, and $B$ a block of the group algebra $kG$ with defect group $D$. If $k$ is sufficiently large (e.g., contains $|G|$-th roots of unity), it is ...
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Let $G$ be a finite group, and let $p$ be a prime. Let $H\subseteq G$ be a subgroup, where $p$ divides $|H|$. We shall say that $H$ is a $p$-local Frobenius complement if $H\cap H^x$ is a $p'$-group ...
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Let $G$ be a finite group. Let $R$ be a discrete valuation ring with residue field $k$, where $k$ has positive characteristic $p$. Let $F$ be the field of fractions of $R$. Let $V$ be a simple $FG$-...
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My first goal is trying to understand the congruence between Fourier coefficients of $\Delta(z)$ and weight 1 modular form $\eta(z) \eta(23z) \pmod{23}$ given by the following: $a(p) \equiv$ \begin{...
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This is a question I will answer myself, as the answer took me long enough to figure out that I found it worth explaining, but it is too advanced and off-topic for my notes and I don't have a blog. ...
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This is so easy to ask that I'm surprised I've never seen it asked before. Let $n\geq0$ be an integer. Let $\left[ n\right] :=\left\{ 1,2,\ldots,n\right\} $. Consider the group algebra $\mathbf{k}\...
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I am reading Swinnerton Dyer's paper on "On $\ell$-adic representations and congruences for coefficients of modular forms". It defines a prime $\ell$ to be exceptional for an eigenform $f \...
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Let $B$ be a $p$-block of a finite group $G$ with respect to an algebraically closed field of characteristic $p$. Suppose that $B$ has an abelian defect group $D$. Let $b$ be a Brauer correspondent of ...
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$\newcommand\sg{\mathbin\#}$Question: Let $k$ be a field of characteristic 2, and let $A$ be a finite-dimensional semi-simple algebra. Let $C_2 = \{1, g\}$ act on $A$ by $k$-linear automorphism. Then ...
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Let $\mathbf{A}\subseteq\mathbf{C}$ be the full ring of algebraic integers, and let $M<\mathbf{A}$ be a maximal ideal containing $p$. Then $k=\mathbf{A}/M$ is (isomorphic to) the algebraic closure ...
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I am studying the Schur-Weyl duality, and I would like to understand the following results: Let $K$ be an algebraically closed field and $V$ be the $n$-dimensional vector space over $K$. Let $Rep_{\...
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$G$ is a finite $p$-group. And $F_{p}$ is field with characteristic p. For $x\in G$, $x-1$ is decompsoable in augmentation ideal of $F_{p}G$ when $x=y^{p}$ for some $y\in G$. $x-1=(y-1)^{p}$.My ...
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Let $f\in S_1(\Gamma_0(N),\chi)$ be a Hecke eigen-cuspform of weight 1. There is a Deligne $\lambda$-adic representation given by $$ \rho_{f,\lambda}: G_{\mathbb Q} \to GL_2(\mathbb K_{f,\lambda}) $$ ...
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Question. Let $G$ be a finite group. Can we find some (commutative) ring $R$ and some positive integer $d=O(\log\lvert G\rvert)$ such that $G$ can be found as a subgroup of $\operatorname{PGL}_d(R)$? ...
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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(\...
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I am interested in understanding the unitary representations of the symmetric group over $\mathbb{F}_{q^2}$. In general, some comments here are relevant Unitary representations of finite groups over ...
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Let $G$ be a semisimple and simply connected algebraic group over an algebraically closed field of characteristic $p>0$, and let $\mathfrak g$ be the Lie algebra of $G$. Let $U_\chi(\mathfrak g)$ ...
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Let $G$ be a finite group and let $k=\mathbb F_p$. Then it is well-known that $G$ has finitely many irreducible modules. However, in general $G$ does not have finitely many indecomposable ...
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In the paper Sur les représentations modulaire de degré 2 de Gal$(\overline{\mathbb{Q}}/\mathbb{Q})$, Serre makes the following comment: Remarque. La relation existant entre "solutions de l'...
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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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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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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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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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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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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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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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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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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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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 $...
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EDIT 21/12: Even if there are no conclusive answers to these questions, I would very much like to know if anyone has noted and attempted to explain the mysterious significance of local subgroups: are ...
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$\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 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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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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For a finite group $G$ there is the Fourier transform $\displaystyle \hat{f}(\rho)=\sum_{g \in G} f(g)\rho(g)$ with inverse $$\displaystyle f(g)=\frac{1}{|G|}\sum_{\rho}d_{\rho}\operatorname{Tr}\left(\...
Jackson Walters's user avatar
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I asked this question on Math Stack Exchange https://math.stackexchange.com/questions/4789924/schur-functors-weyl-functors-in-characteristic-zero, but I got no answers, so I ask the same question here....
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If $\pi_1$ and $\pi_2$ are two smooth admissible representations of $\operatorname{GL}_2(\mathbb{Q}_p)$ over $\overline{\mathbb{F}}_p$ with central characters. I want to prove that if $\pi_1$ has ...
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Let $0\to V_1\to V\to V_1\to0$ be a sequence of representations of $\mathrm{GL}_2(\mathbb{Q}_p)$ over $\overline{\mathbb{F}}_p$, where $V_1$ is irreducible, smooth and admissible. Assume that this ...
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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 ...
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This is an oft-quoted excerpt from John Thompson's article "Finite Non-Solvable Groups": “... the classification of finite simple groups is an exercise in taxonomy. This is obvious to the ...
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In Lusztig's paper "BASE IN EQUIVARIANT K- THEORY Ⅱ", He stated a conjecture about the positivity of coefficients in his famous conjecture. Precisely, (In 9.20) He conjectured that $\tilde{\...
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In general it can be very difficult to compute the invariants of a group acting on a module which is not a vector space over an infinite field. I am interested in the following example, motivated by ...
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This is not a homework or a project question, just me trying to get acquainted to the subject: I am an MSc student who recently came across the Wiles defect numerical criterion (see, for example, ...
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Let $KG$ be the group algebra of a finite group $G$ over a field of characteristic $p$. Question 1: Is there a characterisation when $KG$ is Morita equivalent to a product of local rings? This ...
Mare's user avatar
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Let $G_{\mathbb{Q}}$ be the absolute Galois group of the rationals. I want to know how many isomorphism classes of 2-dimensional, irreducible, pure Galois representations are there over a finite field ...
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It is known that the Steinberg representation $V$ of the group $\mathrm{SL}_n$ over a field $k$ of characteristic $p$ (maybe one needs to assume that $k$ is perfect, I am not sure) is the irreducible ...
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