What is the principal representation?

[Ultraworld]

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.

 

[matt grime]

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?

 

[Ultraworld]

Isaacs: "Characters of degree 1 are called linear characters. In particular, the function 1_G 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.

 

[matt grime]

In what way is that not the definition of the principal (aka trivial) representation?

 

[Ultraworld]

But why don't they just call it trivial representation ?

http://en.wikipedia.org/wiki/Trivial_representation

 

[matt grime]

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.