What is the principal representation?

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Question on representation theory. What is the principal representation? I would like a good clear definition. I can't find it in my book (bad index) nor can I find it on the web.
 
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It could be one of many things. Probably it is the natural representation of whatever kind of object you care about acting on itself by left (or right) multiplication. Where did you come across the phrase if it isn't defined anywhere?
 
Isaacs: "Characters of degree 1 are called linear characters. In particular, the function 1G with constant value 1 on G is a linear F-character. Is is called the principal F-character"

I want to know what a principal representation is.
 
In what way is that not the definition of the principal (aka trivial) representation?
 
I have no idea. Perhaps when Isaacs wrote his book (a long time ago), that was what they called it. It also has links with the principal block too. Some times things have two names. Some times one name has two meanings. That's just the way it crumbles cookie wise.
 
##\textbf{Exercise 10}:## I came across the following solution online: Questions: 1. When the author states in "that ring (not sure if he is referring to ##R## or ##R/\mathfrak{p}##, but I am guessing the later) ##x_n x_{n+1}=0## for all odd $n$ and ##x_{n+1}## is invertible, so that ##x_n=0##" 2. How does ##x_nx_{n+1}=0## implies that ##x_{n+1}## is invertible and ##x_n=0##. I mean if the quotient ring ##R/\mathfrak{p}## is an integral domain, and ##x_{n+1}## is invertible then...
The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...

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