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 ...

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 ...

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 $C_K=\mathbf{C}_G(x_K)$, ...

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 ...

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 $...

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 ...

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 ...

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 $FG$-...

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{...

This is a question I will answer myself, as the answer took me long enough tofigure out that I found it worth explaining, but it is too advanced andoff-topic for my notes and I don't have a blog....

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}\...

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 \...

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 ...

$\newcommand\sg{\mathbin\#}$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 ...

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 ...