5
votes
3answers
539 views

Why are we interested in irreducible representation but not faithful representation?

I am reading some materials of representation theory (of a group). The motivation of representation theory is to represent (by a homomorphism $h: G \to GL(V)$, from the group $G$ to a vector space ...
0
votes
0answers
19 views

Understanding analytic construction of induced representation

I'd like to get some intuition for analytic construction of induced representations as described on Wikipedia. Algebraic construction also described there is much more intuitive and clear to me, but ...
6
votes
1answer
156 views

How to understand $\frac{d}{dt}\{(\exp(tX))_*(Y)\}|_{t=0}=[X,Y]$?

Let $G$ be a Lie group on which $X$ and $Y$ are two vector fields. Let $G\xrightarrow{\exp(tX)} G$ be the (Lie theory) exponential map corresponding to $X$. Then of fundamental importance is ...
1
vote
0answers
56 views

What to take from representation of $S_d$?

I am reading about group representations, and books I read all contain the representation theory for symmetric groups $S_d$. However none of them presents the material in a friendly way. After reading ...
7
votes
2answers
250 views

Algebraic geometry in representation theory?

I heard that today algebraic geometry plays some significant role in representation theory, which is a little surprising because when I learnt representation theory it is basically algebra, topology, ...
5
votes
1answer
340 views

Intuition behind Maschke's theorem

I'm an undergraduate learning about group representations and Young tableaux, and have came across Maschke's theorem stating; If $G$ is a finite group and $F$ is a field who's characteristic does ...
4
votes
4answers
124 views

What is a minimal polynomial of a group element, and why would we care if it was quadratic?

EDIT: the $p$-stable definition I give below is incorrect. I have included the correct definition as an answer to this question. I am trying to understand the definition of a p-stable group. The ...
13
votes
2answers
835 views

Understanding induced representations

Let $G$ be a group and $H$ be a subgroup. Let $\phi:H\rightarrow GL(V)$ be a representation of $H$. There are three constructions in Wikipedia, but I am not really convinced by these. My question is: ...
5
votes
1answer
55 views

Something behind the substitution $h^0=\frac{1}{|G|}\sum_{t\in G}\rho^2_{t^{-1}}h\rho^2_{t}$?

I am quite new to representation theory and I reading Serre's Linear Representation of Finite Groups. In the first and second chapter, one trick he uses quite often is the substitution ...
3
votes
1answer
194 views

Intuition on the definition of “rational maps”

I'm studying some representation theory on $S_n$ and $GL(V)$ and tensor spaces, and have come across a lot of material involving rational representations. I'm not really an algebraic geometer by ...
45
votes
3answers
902 views

How to think of the group ring as a Hopf algebra?

Given a finite group $G$ and a field $K$, one can form the group ring $K[G]$ as the free vector space on $G$ with the obvious multiplication. This is very useful when studying the representation ...
17
votes
2answers
538 views

Categorical description of algebraic structures

There is a well-known description of a group as "a category with one object in which all morphisms are invertible." As I understand it, the Yoneda Lemma applied to such a category is simply a ...