Undergrad What does V^G mean in linear algebra?

[cask1]

TL;DR

Hi. Newbee question. What does the notation V^G mean where V is a vector space and G is a group? I found it in: A linear algebraic group G is called linearly reductive if for every rational representation V and every v in V^G \ {0}, there exists a linear invariant function f in (V^*)^G such that f(v)<>0. I can't find a definition of the notation using google. Thanks.

Hi. Newbee question. What does the notation V^G mean where V is a vector space and G is a group? I found it in: A linear algebraic group G is called linearly reductive if for every rational representation V and every v in V^G \ {0}, there exists a linear invariant function f in (V^*)^G such that f(v)<>0. I can't find a definition of the notation using google. Thanks.

 

[Office_Shredder]

Normally for sets $X$ and $Y$, $X^Y$ is the set of functions from $Y$ to $X$. In a context like this, it's often meant to be limited to functions of the desired type, e.g. only representations.

That doesn't really make sense here, since a representation is a map $G\to GL(V)$, not a map from $G$ to $V$.

 

[Infrared]

When a group $G$ acts on a set $X$ (in your case $V$), sometimes one writes $X^G$ for the set of invariant elements, that is, the elements of $X$ that are fixed by all group elements. Note that if $G$ acts linearly on $V$, then it also does on $V^*$, so it makes sense to write $(V^*)^G$ in your example.

 

[mathwonk]

The AI summary explanation that ends this thread, visible only when not logged in, incorrectly equates V^G with V ^*, and hence virtually every statement in it is false.