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# How do I determine the smallest dimension of an irreducible $\mathbb{F}_p[G]$-module with a prescribed trivial fixed point space?

This is a crosspost from MSE since I haven't found an answer there yet.

I am not very familiar with modular representation theory or Brauer theory yet, however lately I have needed to use $\mathbb{F}_p[G]$-modules for constructing counterexamples. Right now I am just making them in Magma, but I am looking to understand what I'm doing a little better.

Say that I am given a group $G$ which I would like to act on an elementary abelian $p$-group $E$ so that a prescribed proper subgroup $H\leqslant G$ acts fixed point freely on $E$. E$, with$|H|$and$p$coprime. So, I construct an absolutely irreducible$\mathbb{F}_p[G]$-module with a trivial fixed point space under restriction to$H$. Often I have other subgroups which I would explicitly like to not act fixed point freely on$E$(usually faithfully or trivially) in which case I also insist the module have a nontrivial fixed point space under restriction to those subgroups. I am trying to figure out how to determine the smallest dimension of such a module - in other words, I want to know how big$E$has to be. Obviously this is going to depend on the structure of$G$, and in particular the structure of$H$. Which group properties determine this, precisely? Does anyone know where I could read about this specifically? There is some existing theory. For example, one can prove that if$F$is a Frobenius group with cyclic complement$C$, and$F$acts faithfully on an elementary abelian$p$-group$E$, then$|C|$divides the minimum dimension of$E$over$\mathbb{F}_p$. Are there more general results? 3 added 11 characters in body This is a crosspost from MSE since I haven't found an answer there yet. I am not very familiar with modular representation theory or Brauer theory yet, however lately I have needed to use$\mathbb{F}_p[G]$-modules for constructing counterexamples. Right now I am just making them in Magma, but I am looking to understand what I'm doing a little better. Say that I am given a group$G$which I would like to act on an elementary abelian$p$-group$E$so that a prescribed proper subgroup$H$H\leqslant G$ acts fixed point freely on $E$. So, I construct an absolutely irreducible $\mathbb{F}_p[G]$-module with a trivial fixed point space under restriction to $H$. Often I have other subgroups which I would explicitly like to not act fixed point freely on $E$ (usually faithfully or trivially) in which case I also insist the module have a nontrivial fixed point space under restriction to those subgroups.

I am trying to figure out how to determine the smallest dimension of such a module - in other words, I want to know how big $E$ has to be. Obviously this is going to depend on the structure of $G$, and in particular the structure of $H$. Which group properties determine this, precisely? Does anyone know where I could read about this specifically?

There is some existing theory. For example, one can prove that if $F$ is a Frobenius group with cyclic complement $C$, and $F$ acts faithfully on an elementary abelian $p$-group $E$, then $|C|$ divides the minimum dimension of $E$ over $\mathbb{F}_p$. Are there more general results?

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# WhatdeterminesHowdoIdetermine the smallest dimension of an irreducible $\mathbb{F}_p[G]$-module with a prescribed trivial fixed point space?

This is a crosspost from MSE since I haven't found an answer there yet.

I am not very familiar with modular representation theory or Brauer theory yet, however lately I have needed to use $\mathbb{F}_p[G]$-modules for constructing counterxamplescounterexamples. Right now I am just making these them in Magma, but I am looking to understand what I'm doing a little better.

Say that I am given a group $G$ which I would like to act on an elementary abelian $p$-group $E$ so that a prescribed proper subgroup $H$ acts fixed point freely on $E$. So, I construct an absolutely irreducible $\mathbb{F}_p[G]$-module with a trivial fixed point space under restriction to $H$. Often I have other subgroups which I would explicitly like to not act fixed point freely on $E$ (usually faithfully or trivially) in which case I also insist the module have a nontrivial fixed point space under restriction to those subgroups.

I am trying to figure out how to determine the smallest dimension of such a module - in other words, I want to know how big $E$ has to be. Obviously this is going ot to depend on the structure of $G$, and in particular the structure of $H$. Which group properties determine this, precisely? Does anyone know where I could read about this specifically?

There is some existing theory. For example, one can prove that if $F$ is a Frobenius group with cyclic complement $C$, and $F$ acts faithfully on an elementary abelian $p$-group $E$, then $|C|$ divides the minimum dimension of $E$ over $\mathbb{F}_p$. Are there more general results?

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