3
$\begingroup$

What is the exact definition of The General Linear Group Representation of a group $G$? I know that a linear group representation of $G$ is a homomorphism $\gamma$ from $G$ to $GL(n,F)$, where $GL(n,F)$ is the group of all invertible $n$ by $n$ matrices with elements in the field $F$. I don't understand what is the general linear group representation of $G$. At following I wrote the exact sentences of the paper I'm reading, I think it is false because the usage of "the" instead "a". "a $G$-invariant multivariate polynomial over reals is a multivariate polynomial $f$ with $n$ variables $x_1$,...,$x_n$ such that, for every $g$ in $G$ we have $f(x)$=$f(\gamma (g)x)$, where $\gamma$ is the general linear group representation of $G$, i,e, $\gamma$ is a homomorphism from $G$ to $GL(n,k)$". (k is undefined but I guess that should be $R$)

  • 0
    Who says "general linear group representation"? – 2017-02-08
  • 0
    The letters $GL$ stand for "General Linear", and one reads $GL(n,F)$ as the "general linear group of $n \times n$ matrices over $F$". Notice that there are two parameters: the rank $n$, and the field $F$. So, "the" general linear representation of $G$ does not make sense. What would make sense instead is "a" general linear representation of $G$ of rank $n$ over a field $F$, which is just what you say, namely a homomorphism $G \mapsto GL(n,F)$. But then the common usage is to kind of drop the word "general". – 2017-02-08
  • 0
    I have edited my question to make more clear. – 2017-02-08

1 Answers 1

3

For most people, unless specified otherwise, the word "representation" means the same thing as "linear representation," which is just what you wrote. If someone said "general linear representation," this would be a little unusual, but would mean the same thing.

I suspect that you are reading a text that wants to contrast "linear representations" (maps $G \to \text{GL}_n(F)$) with "projective linear representations" (maps $G \to \text{PGL}_n(F)$), but it's hard to know what they mean without seeing your source.

(Pedantic note: a better way to define a representation is as a map to $\text{GL}(V)$ where $V$ is some vector space; this is the same as your definition but doesn't require a choice of basis. Also your definition forces the representation-space to be finite dimensional, which you may or may not wish to assume depending on what area of math you are in.)

EDIT: it appears your author has some fixed representation of their group $G$ in mind. Obviously, their definition of invariant polynomial depends on that choice, and they should have phrased this better. Perhaps there is some choice of representation made earlier in the paper?

  • 0
    I have edited my question to make more clear. – 2017-02-08
  • 0
    I see. Well it's not true that every $G$ has some canonical representation, so I'd say this is not great use of language by the author you're quoting, unless this is better-explained somewhere else in the paper. – 2017-02-08
  • 0
    Thank you @ hunter. Yes, I think this is a wrong use of language by author. – 2017-02-08