hello :D

hello

Well Anon, I did solve first 2 problems that I need to solve =P

mostly thanks to you :D, that first problem I sent you

about using schur lemma,

Did not give you all detail of it, and also it might be not well written

ill send you all detail, and if you feel something is off tell me so i know what i would do :D

okay

ill send youy both screen shots now

I have to be done with these questions in like a week >< so at least I got the time =p i wish we could go over some material we did in class ,that I did not feel so secure about ._.

so to learn rep theory properly , how much knoledge one needs on rings ,fields, modules

I want to understand field extention and module, if you could help me with that , that would be super awesome :D

i talk alot I know ><

okay

just to avoid potential comfusion, my teacher uses transpose of the left ^t A

sorta like that >< anyway :)

so, with that question, you need to know the matrix of the dual rep is $\rho(g^{-1})^T$, and so $\rho$ isomorphic to its dual means $\rho(g^{-1})^T=A\rho(g)A^{-1}$, which is equivalent to $\rho(g)$ preserving the bilinear form $x^TAy$

then you need to argue $A$ defines a symmetric or symplectic form I guess? not sure where irreducibility comes into play

I honestly dont know yet, since I did not read that far

But that question is not why I wanted to talk to you :D we can take it later when am ready for it =p

I got many defintions /concepts i want help with :D

the major two are Modules and field extention ><

tensor Product also ._.

okay. what about them?

well the thing is , we did not do any of those on AA course

but to take rep theory , they assume we know em

so those who took galois course or other things of that nature had no problem

but in my uni , in order to be accepted to take rep theory, they only ask for linear algebra and AA

a field extension L/K is when L is a field that contains K, i.e. it has more numbers than K has (in the case that L and K contain numbers)

hmm

So I saw stuff like Sqrt(5) or Sqrt(2)

some people say a module is like a vector space over a ring, but this can be a counterproductive intuition - modules can fail to have bases, have bases of different sizes, multiplying by nonzero scalars can still give zero, etc. I think of it as a "linearization" of a group action, like G acting on X yields the group algebra C[G] acting on the free vector space CX

from what you said R/Q , C/R make sense?

yes, R/Q and C/R are field extensions

Hmm can you explain with example? and more basic stuff :D

explain modules?

this is my first encouter to those topics :D

field extension did not get it yet too

So we write R mod Q

no, don't write R mod Q

that's a different thing, though same notation

isint that symbol modding out?

ha

aha*

well I need to look for some videos on these stuff :D

Q(sqrt2)/Q is another field extension

Q(sqrt2) is a field, and it contains Q

so we "extended it from Q by addion 1 element?

adjoining sqrt2, yes

of what purpose?

eh?

Okay sorry am asking bad questions because I dont know anything about these ._.

my question was adding sqrt(2) , what did we gain

more numbers

not everything is a rational number

hmm

okay lets go to modules :D

like Little intro so I dont get lost while studying it alone

R is a ring, M is an additive group, R acts on M

(r+s)m=rm+sm etc.

one don't need intuition of these things right?

just accept the rules and work with the structure of those things?

one gains intuition

okay that really was helpfull :D

I allways Think that I need to have intuition Before doing it

If F is a field, then F^n is a coordinate vector space over F. Similarly, Z^n is a Z-module (n-tuples with integer entries). And Z/mZ is also a Z-module. And C[x]/(f(x)), the quotient ring, is a C[x]-module, because you can multiply stuff in C[x]/(f(x)) by things in C[x]

Okay thanks alot anon :D, i Think what I need to do is , properly read About these topics, now that i have a small idea, ill come back to what you said after gaining some experience :D

You are the best anon :D

np

Btw I never asked, what Courses do you take now ?

I assume you are proffesor right?

students take courses, professors teach courses

I'm a student-teacher

Nice :D, I also want to continue learn other stuff, even when i have a job =p

Okay , Thanks alot again ! ill come up with better questions next time:)

I'm doing an algebraic geometry seminar thing with a couple other people, but otherwise doing math on my own, leading group activities with others (putnam group, problem-solving group, etc.) and teaching algebra / trig / calc

nice :D

are any avaible?

lecture notes / videos anything ? :)

I did compete once in mathematical olympiade in sweden

got 27 /50 haha

most of my stuff is on paper, many unrealized plans for lecture notes

it's hard to get typed notes to match the organizational pedagogy of written ones

I understand =p but if you once plan to have lectures on videos , pls send me a link :D

hmm, okay

Okay now I need to do my part of the work !