I'm having a brainfart. Why is every irrep a summand of the reg rep? Notes say it's cause the former will be a homomorphic image of the latter, but I fail to justify this.
and given the regular representation has a default construction, I expect one would have to prove the universal property rather than take it as a given, so I'd be back to where I was
Well, that is a non-obvious fact. But it is easy to show that every irreducible representation is the homomorphic image of the regular representation, assuming that we are in the situation where we have complete reducibility.
Let $R$ be a ring (with unit, not necessarily commutative, blah blah). $M$ is a free $R$-module generated by $S \subseteq M$ if and only if for every $R$-module $N$ and every map of sets $S \to N$ there is a unique $R$-module homomorphism $M \to N$ extending $S \to N$.
@BenjaLim Yeah. But why is that so surprising? $\mathbb{R}^n$ has lots of wiggle-room, and you have deleted only finitely many points.
okay, for every mapping $\{1\}\to M$ with $M$ a $kG$-module, we can extend by linearity to get a mapping $kG\to M$. the space of mappings $\{1\}\to M$ is iso to $M$ itself, and by extension we get a mapping $kG\to M$ that is surjective. right?
Introductory representation theory is usually done with $G$ finite, $\operatorname{char} k = 0$, and $k$ algebraically closed. But then you may as well take $k = \mathbb{C}$...
I think, to make Schur's lemma and Maschke's theorem go through, it is enough to take $G$ finite, $k$ algebraically closed with $\operatorname{char} k$ not dividing the order of $G$.
right actions in general behave like exponentiation. (in fact I have a group theory text that writes $g^\alpha$ for $g\in G$ and $\alpha\in\mathrm{Aut}(G)$.)
(it writes automorphisms from the right for some reason)
I think of exponentiation as being characterised by these equations: $$m^{a_1 + a_2} = m^{a_1} \cdot m^{a_2}$$ $$m^{a_1 a_2} = (m^{a_1})^{a_2}$$ $$m^1 = m$$ $$m^0 = 1$$
we could assume a commutative ring $S$ has the property that $S^\times$ is a right $R$-module, and speak of an extension of this to the rest of $S$ (so like a module acting on a monoid instead of a group, and the additive structure takes the sidelines)
Ultimately though, what I'm looking for is a non-commutative version of exponentiation. The only one I know of is the ordinal numbers, but that's still "numbers"...
@MattN. Na, I don't think that is true. However, if you are able to work for 8 hours it will only do more harm than good. It can be very useful if you say can only do that for 10 minutes. Don't take Erdös as an example, he's a crazy man.
The nice thing about hosted blogs is that you can just forget about it for a few years and it will still be there when you come back. Without having been hacked and defaced...
They don't even think about paying me business class. They just laugh 8-(.
@DanBrumleve Perhaps if you read some substantial papers I'm sure you will not be able to read more than 2-3 on any flight (before you can access the internet)...
@JonasTeuwen I wouldn't : ) And I'm quite surprised that I can do maths for so many hours. I knew I could write code for say 10 hours (-ish) but that's much less intense. I guess with the right sort of motivation one can do lots of things one would not normally be capable of.
@MattN. It is a very unusual skill. Most people certainly cannot do that (well, not good anyway). Amphetamines would totally fuq it up. You'd be jumping around instead.
@OldJohn Heh... 8-).
@MattN. They seem to work by fixing some understimulation instead of adding to already good stimulation... Then you will be very non-productive. Many ideas, write nothing down would be the best option.
Usually, after 2-3 hours of coding, I'd like to vomit. So no 10 hours for me.
@JonasTeuwen It's like ETH: I'm full of energy and motivation and interest by nature and then ETH is like sticking a pipe down your throat and pumping amphetamines into you. --> zero productivity x_x --> death (eventually)
@DanBrumleve I believe the biggest cause really is not systemic. For one it is quite acidic, secondly dry mouth and third grinding teeth. Finally poor oral hygiene. You'd don't see it that bad in pharmacotherapeutic use.
i used to value the perceived productivity increase but now i tune into a similar benefit in my own adhd itself even though i can't control it as easily.
actually for me it was more of a physical thing (hyperactivity) than not wanting to read boring shit (which i won't tolerate either although it's not as bad as staying in one place).
while i'm here i guess i should promote my latest question math.stackexchange.com/questions/179360/… which yesterday i was afraid of getting closed but i got some good feedback and information from @joriki
fwiw, going on ritalin was really awesome at age 13... it was so much easier to tune out the class and work on software in my notebook. but it got problematic later and i didn't totally quit speed until my early 20s.
(switched from ritalin to dex at some point but it was pretty much the same thing.)
