Maschke's Theorem states that every representation of a finite group is completely reducible, meaning it can be decomposed into a direct sum of irreducible representations. This theorem is analogous to the fundamental theorem of arithmetic, where every integer greater than 1 can be uniquely factorized into a product of prime numbers. The proof relies on the fact that every representation of a finite group is equivalent to a unitary representation, and every unitary representation is either irreducible or decomposable. The complete reducibility is proven by induction on the dimension of the vector space, where the base case is one-dimensional representations (which are always irreducible), and the inductive step shows that if a representation is decomposable, it can be written as a direct sum of two proper subrepresentations, each of which is completely reducible by the induction hypothesis.
Day 129 – Practicing Math Live – Ch. 3 Group Representations
Added:Hey, what's going on YouTube? I believe that we're live now. What's up? We are back. Um, another day practicing mathematics. I will be working on representation theory of finite groups.
This is a beautiful subject. Um, uh, it's technically graduate level, but I think that I I've been rethinking how I want to structure all of my streams.
So, this is the primary focus of this week. it will be a graduate level topic.
But I think that what I can do is I can interleave other levels of the same subject um and also work some problems from undergraduate books or even hypothetically I think some of these books a high schooler could learn. I think that you could learn this topic at different levels and I could be mixing the different levels at the same time throughout the week. So, we may start adding in a little bit of work from Dumbit and Foot um as well as a little bit of work oops a little bit of work from Gallion. So, generally speaking, this week is algebra week. I'll be working abstract algebra. Uh, and this is the primary focus, but I will be pulling in a little bit of material.
Man, I have so many of these freaking algebra books, don't I? Here's that Here's that same guy, Serge Lang. That scary ass dude. His algebra book is hardcore.
So, dude, look at all these algebra books that I How many algebra books do I have? Um, Dude, I'm like, what the hell?
Okay, this is getting a little silly, dude. This is actually getting kind of silly. Look at all of these algebra books that I have, bro. Like, [laughter] a lot of these are just the same exact topic, actually. Like, abstract algebra at different levels. So, this week is algebra week. Next week, we're back to basic math week. Week after that, number theory week. I I think that that's my new structure, my new idea. Um, and not only work through one book, but work through multiple books, work through exercises from multiple books, uh, with one of them being the focus, one of them being like the primary target, and then we've got a bunch of supplementary books as well that won't be working. Dude, this is crazy. Let me just see this for a second.
I've never I don't know if I've ever done this.
Look at all these freaking books, dude.
And I know that there's more. Um where is Yeah.
Here, this one. I mean, can I call linear algebra part of this series, too?
Linear algebra technically, I think, qualifies as part of this, too, doesn't it?
It does. I mean, it really, really does.
Like linear algebra is so much more abstract algebra. It's the same thing than number theory. Like, it's not really number theory at all. Um, bro, look at this craziness.
[laughter] Dude, you can't even see off camera like the bottom of this stack. Oh my god, dude. Um. Oh, I need this one on top of everything else.
Oh, it's heavy. Oh, dude, it's heavy. I need to do some more push-ups, bro. What's up, Sugash?
So this is algebra week and we will be working algebra at different levels primarily focusing on this one on this one specific topic but I've got all these other books available to uh to supplement and do a little bit across all of them.
Oh, look at that. Look at that beautiful stack, dude. [laughter] Look at this beautiful stack. You can't even see the bottom of it, but there's so many freaking books in this pile.
[sighs] Let's see. Can we get that pretty here?
We're gonna We're gonna do this.
Some of the scariest books are tiny.
Some of the really scary books are these tiny books on top. Um, speaking of Where is I I thought I just grabbed it. Where did I Oh, it's down here. There's a tiny book down here. Do you have any advice for undergrads?
Um, just learn to work, man. Like, learn to work your ass off. uh in undergrad you should be learning how to work um I don't know learn about deliberate practice I think hi mate do you have any advice to learning math again at 25 years old um follow my course dude honestly follow my course on basic mathematics and if you're learning math again learn it the learn it the right way. Dude, Lang and Eisenbud are a close are are neck andneck. I think Lang is just a little meteor than Eisenbud.
Eisenbud is at the bottom commutative algebra. Um but Lang and Eisenbud are the two beefy the two beefy boys in this stack. Um so let me see Rizzle. Um, follow my basic math series.
Um, here's a playlist for that. Uh, yeah. I study advanced math and the tiniest books are the worst. The yellow and white ones, but they are wonderful. Yeah. Yeah. The tiny books are the scary ones. You won't you I mean you should be afraid of the of the big beefy boys too, but like the tiny ones are the scariest ones.
I don't know which white book you're thinking of, but this one, it doesn't belong in this algebra stack. It belongs in my analysis stack. But this one, bro, is uh this one is scary as hell. And it is tiny. It is one of these tiny scary books. Like the tiny ones you you should be afraid of.
You think Gallion is a horrible book? I just started reading it. I I like it. I think Gallion's nice so far. Um, I think I think Gallian's nice. Why do you Why do you think Gallion is horrible?
Interesting.
Hi, thank you for your time. Thank you for tuning in. Thank you. Thank you for being here. Um, linear algebra is not quite the focus as much anymore. Um, so I kind of just want it to be supplemental material.
The real focus nowadays I want to be these algebra books.
10 likes equals 10 push-ups, everybody.
So, hit the like button and I'll do some push-ups. I think that's going to help my classical guitar.
And apparently, it's also going to help my mathematics as well for the times that I need to show you all a stack of scary books.
Uh, thanks mate. Really appreciate it.
Yeah. Yeah, dude. Thank you for being here.
Okay. So, um we will be this week working algebra. Um and the primary book is this top one.
This primary book I am going to be focusing on and then everything else in the stack is kind of supplemental uh for the sake of learning learning this subject. Um, but what's new about this structure is that we will be working some simpler undergraduate or even high school level math at the same time as everything else. Um, and that reminds me that I should probably add these two to this stack as well.
Challenging problems in algebra. This looks like a nice book and it looks like it's kind of like high school like competition like imo style problems in with high school algebra. Like I kind of want to work problems from that book as well. But this week is algebra week.
You need a number theory book. So yeah.
Uh that's the thing. I do have number theory books and I'm that's going to be a different week. Uh that's a different stack. [laughter] Uh yeah. Yeah. Number theory. So right now it's uh one week of high school math, basic mathematics by Lang and then the next week will be number theory and then the week after that will be algebra primarily representation theory. Um but yes, I do have another stack of number theory books and we will focus on those.
I think you have to read before you start with these. I mean I know number theory. I know some number theory, but um number theory is getting its own entire week, I think.
Uh I just realized at age 25 that you really love math. Is it too late? No.
Absolutely. Absolutely not. And if you want to try switching careers to become an AI engineer. Interesting. So I think that you should absolutely learn math.
If you want to become an AI engineer, you need to learn to love math.
Now, not all of this will be useful to become an AI engineer, but it a lot of it will a lot of it will be useful because you need to learn abstract higher level math in order to uh to be able to work on AI. Like AI is like so much of AI is linear algebra. Like you need to be really good at your linear algebra. Um, yeah. I mean, you could absolutely learn. You could absolutely catch up if if that's what you want to do. Uh, but like it's never too late is like really the I think the unique new thing, the the thesis of my channel. And of course, I'm not the first person to say it, but I feel like I'm the first YouTuber to focus on this this much.
It's never too late, dude. You can pick up math as a hobby just because you like it and it's good for your brain and it will help you in your career, whatever your career is, cuz it'll make you sharper, dude. It'll make your brain work overtime because math is hard and math is good for your brain and it'll be good for your job, whatever your job is, I think. Um, and you can do that at any point. It's never too late to start learning math. Never too late. Um, and AI will probably be working on itself in the coming days. Yeah. Yeah. Uh, yes. I don't want to go down that rabbit hole, but yeah, Skittles are bad. I think Skittles is is on to something there.
Um, I don't know, dude. I mean, the way that AI is moving, we might all just become cats entertaining ourselves, you know?
>> [laughter] >> like we might just become like Luna and just not like I don't understand the math that we're doing here, but I just like to to hang out and cuddle and clean myself and entertain myself.
Um, but that's okay. We can entertain ourselves. We can study math for fun.
Like math is still good for you, dude.
Um, so you think Gelfind is better than laying book for algebra foundations?
Yeah, I thought about picking up Gelfind. That's That's another one. I'm down to push stuff off the table. Yeah, like Luna does. I want to see her knock this stack over. Um yeah, like for real.
Um I don't know. Uh I thought about picking up Gelfind. I was researching books. I just really like Lang's writing. So, I just trust because I tried working through Big Boy Scary Lang algebra. Um, I just trusted that his book would be good for high school math and I'm I'm enjoying it so far. Best work is usually done in the early 20s though, statistically speaking. Yeah, I mean it may be too late to become the next Terrence Towel. Okay, like it may be too late for you. I'm sorry. Let's just accept that and that's okay. You may not become a worldclass top 10 mathematician in the world because to do that you may it's possible that you needed to start before 10. But dude, I just think that learning math is just a decently good way to spend your time. I don't think anything is just objectively a better way to spend your time. Like I'm not saying that it's like some morally better way to spend your time, but like I don't know, dude. Like I just personally speaking, I feel more fulfilled when I study math as a hobby than when I scroll TikTok as a hobby. Um I don't think math makes you more intelligent like incremental and fluid intelligence, but it does give you new perspective/view for approaching problems and you gain valuable skills.
Yeah. Like I think math is good at gaining concrete valuable skills including confidence. Like building confidence honestly like I've solved hard problems before. They were really hard. I was stuck. I was mad cuz I couldn't solve that problem and then eventually I got through that problem and and it was a eureka light bulb moment. I got through hard problems before. I can get through this hard problem now. like that attitude and mindset shift I think is something that math is really good at. Um you know just to just to gain confidence in your problem solving abilities or just trusting you know that you that there is an answer to this thing and that you can you can work through some hard problem and you can enjoy it in the process. Um, uh, somebody asked if I'm going to do, uh, algebraic geometry. Yeah, someday, dude. Someday. Um, it's at the bottom of this [ __ ] stack, dude. Look at this.
All right. At the bottom of this, we got Eisenbud. Commutative algebra with a view toward algebraic geometry. We will get there. We will get there someday.
But dude, this is a lot of this is a lot of math to keep myself entertained with.
Still trying to fully understand quians.
I barely understand quattonians myself.
I mean, quatians do create an interesting number field or interesting number system.
Um, and that's about all I know about it. I know them as an example of a group, but I don't know I don't know very much beyond that. You don't need to fully understand quatnians to study math. Um, there are places that you can apply quatians if that's what you're interested in. Like I think that they're quite useful for game development and some other things, but nice say stuff. One of the most fulfilling things was figuring out why R and R cross R have the same cardality.
Neat.
Yeah. Yeah. Math is just I don't know, man. I just I do think math is objectively quite good for your brain.
Like I don't I'm not going to say that it's the single best way to develop fluid intelligence, but I think it I think it keeps your mind sharp. I think that it is better than Tik Tok for keeping your mind sharp.
And yeah, I don't know, man.
Yeah, I'm I'm going to need some help. I am definitely going to need some help to read all these books. And I there's even more that just don't fit right now.
We also have these. So, this is algebra week. These are all the books right now as well as of course some every now and then I'll be mixing in some examples from basic math a little bit but not not very much not very much but we will get a little bit of basic math. Um hey watching your videos for a week or so happy to catch a live I love math and do some every day. That's awesome.
Christina, I'm very happy to hear that.
And you're a 3D artist. Yeah. So, uh, don't need to understand it. Just use it to fix gimbal lock. I've heard that. I've heard of that term before.
Um, absolutely. Like, mathematicians, we don't really use quatnians very often, I don't think. Um, I mean, they are an interesting number system, uh, but you don't it's not like you need to fully understand them. I I'll I'll probably do a video about Quattonians at some point.
Um, okay. So, speaking of videos, I do want to give a shout out that uh I just uploaded episode 4. Dude, I just uploaded episode 4 in my series of basic mathematics. This week we are doing um this week we are doing multiplication.
We just introduced multiplication this week. So go check that out. This video is a theory video and then later this week I will get another video on examples of multiplication and including some interesting examples that you probably haven't seen. Uh it's just so interesting. I need to understand that.
Right. Right. That's how I feel too.
Permutations and combinations. Yeah, dude. There's a lot of that stuff. Um which book in the middle are you talking about? Big boy Lang. Big boiling is is scary. This this one is scary. Is this the one that you're talking about? Um I'm trying to find actually find a hobby instead of just filling time with YouTube between things I have to do. I do play games, but that that has also been boring, so it might be time to do math. Yeah, dude. Do you remember like in the pandemic chess just out of nowhere started kind of springing up as like a thing that all the streamers were doing? Bro, can we get like freaking Lud in this in this chat? Can we can we get like Can we get the big streamers with like a million followers plus like Lud and Speed and uh Kai and freaking all of them, you know? Like can we get Pokemon in the chat doing math? Like why not?
Like people like chess became big, dude.
Chess and math are not that different.
Like it's just fun. It's just a stupid fun way to use your brain and spend some time, you know? Like, dude, you are not going to be better than AI. AI is already better than all of us. Well, that's not really true, but AI is getting really damn good at math really, really fast. Yeah, dude. Why not? Why Why couldn't we get three blue, one brown in here? up in here, dude. Like, we could like he might be lurking at this point. Like, I would be shocked, dude, if YouTube has not recommended that first video that went viral. I would be freaking shocked if three blue, one brown, if uh Grant hasn't seen that video yet or like at least been offered that video. I would be very surprised.
Um, that video now has 3.2 million impressions. YouTube has offered that video 3.2 million times. That's so much.
Uh, so that's crazy. Uh, AI is artificial. Yeah, it takes all the humanity and fun out of it. Um, but like all of this was invented by humans, dude, over thousands of years. And all of these arguments are still going to be beautiful and still just timeless. Like all of this was made by humans. This was human math before the machines took over. And it it will take a lifetime, you know, for a lot of people to understand this stack of books. And dude, like it's just a good way to spend some time. Like it like this is going to keep me busy for a while. I'm never going to understand everything in this stack of books, but that's that's okay.
That's I think that that's a good way to spend some time. Um, okay. So, somebody said somebody out there notified me that we just hit the magic number. We got 10 likes. 10 likes equals 10 push-ups. So, let's do it. 10 push-ups. I don't have a push-up cam yet, but thank you everybody. Um, I talked with my guitar teacher Chris yesterday and he said to work on my posture, I should do more push-ups. And I was like, funny enough, dude. Funny that you would say that. Um, so 10 push-ups. Thank you for 10 likes.
All right. Hell yeah, dude.
[snorts] Um, you all think you all might think that I'm joking about getting a streamer or like somebody like Lud. Uh, I will offer like Lud gets free access to the weekend Zoom group. There's currently three people in this weekend Zoom discussion group.
Like, of course, Lud can join for free for freebies. Uh, so go tag him. go like go tell him, you know, in the chat to check out JC practices math and that I want to teach Lwig or Speed or Pokemon or Kai or any of them. Uh, I would love to teach them math. I think that'd be freaking sick.
That's so awesome that you're working on your posture. Yeah, because of classical guitar.
Uh, for those who don't know, this is not my first rodeo. This is actually my second stream that I've created, my primary one. Uh, I see at least one guitar regular in the chat. Um, I do classical guitar, also a wonderful deep hobby. And uh, yeah, I'm studying classical guitar at a high level, at kind of a semi-professional level. And yesterday my teacher was like, "Yeah, dude. You should you should do more push-ups." I'm like, "Fine. All right. Okay. Fine.
Okay. Uh, so let's We're in algebra week this week. Thank you everybody for being here." Uh, you asked nature and stars are nature. What?
Oh, you all are talking about astronomy.
What are you talking about?
Yeah. I think humans like the process and feeling of learning and doing art/math.
Even if AI can do it better. Exact.
That's my point. It's like even if even if the freaking robots are better than me at this stuff, I still want to learn math anyways. [laughter] Like maybe I'm not going to be able to invent new math, but I can still learn it for myself. And in the process, I think math will have a good effect on the rest of my life everywhere outside.
Um, there are a few of us that love math in the card making world. When you say card making, are you talking about like tabletop games? Like are you talking about Magic or Netrunner or like one of these like card games? Like dude, there's so much like it's so mathy. Like I've only played one and gotten really into one card game called Android Netrunner. Just such a beautiful game.
Like all the mathematicians I knew were like the biggest like game nerds. They all love games cuz math is just like a game. It's just like a tabletop game, dude. Like for real. Um is that what you're talking about?
Oh, paper crafting. Okay. Like greeting cards. Okay, that's cool. That's cool. I don't I don't know the connection there, but I believe it, dude. Like, math is just [sighs] I just love it. I just I think it's really pretty. I think it's really nice. Um there's a funny there's a really nice quote and I talk about this in episode 4 of my basic math series.
There's this wonderful quote that just made me smile and chuckle and I got so happy when I heard this uh when I read this in the cafe preparing you know for my upcoming videos. The following three formulas are used constantly. They are so important that they should be thoroughly memorized by reading them out loud and repeating them like a poem to get an oral memory of them. Like Lang is just like dude this is poetry like have you ever thought of those formulas? A + b 2 is a 2 + 2 a + b. Have you ever thought of that as poetry? It is. It's poetry. That's It's incredible.
Like there's just a side this romantic beautiful side of math that is just not represented, I think, in in the high school classroom, at least in the US.
Like I I didn't see this side of math until I went to college.
Um, so yeah, I do think that there's some poetry in it all. Um, and you always enjoy the visual aspect of math.
Um, great. This stuff has some really lovely visual visual side of math. This is this week we're studying uh the math about symmetry. This is actually the math of symmetry and it's beautiful stuff. So, we're going to um start locking in. By the way, I used to always just have black notebooks. Like I just loved my black Moleskine notebooks and just very consistent. Loved them so much.
Nowadays, I'm trying to mix it up. I've got color matching notebooks for every subject. I'm working like three books.
Like seriously, right now.
And dude, like it's just nice. Like it's just nice that the the notebooks match.
Like dude, look how cute that is. Look how cute that is. Like this yellow notebook matches this yellow book. This orange notebook matches this orange yellow book. And I've got like a green one that matches my green book.
Yeah. So, whatever inspires you just a little bit more to just keep studying, keep working. Like for you, if it's if it's having like a cute color matching notebook, like that might that might just help a little bit. Um, I have 10 kids and homeschool. You have Wait, did you When you say that you have 10 kids, like you you had 10 kids, like dude, that's a lot of kids.
Good for you. Wow. Uh, as well as tutor others to understand math. Are you saying that you're Yeah, you have 10 kids and homeschooled them all. Wow, that's awesome. Um, I I want to I wish I could homeschool. I just I think that there's challenges in that, right? Um, it can be fun and is around them every day. Math is just it's just everything, dude. Math is like [sighs] it's just so pretty. So pretty. And it's good for your mind. That is a lot. And you're all yours. Congratulations.
That is amazing.
Like that's that's incredible. Um wow.
Wow. Um [snorts] Okay, cool. So, let's let's get going. Good for you. Wow.
Uh uh so thank you for hanging out during this 30 minute introduction.
Uh welcome back to algebra week. Today is day two in algebra week. Um what's biology like in abroad? What is biological? What is not? How do we define that? Interesting.
Hey. Hey. What's up, Arad? I'm going to do this one last time just for Arod.
He's like a he's a regular here on this channel. Arod, I'm I'm changing up the structure just slightly.
Uh, instead of this week being representation theory week, this is still the focus. This is still the primary book, but instead of this week being representation theory week, this is algebra week. So all week we are going to be working algebra broadly defined.
Um this is good for my arms, dude.
Welcome everybody to algebra week primarily focused on representation theory of finite groups but this is algebra week this week uh hell yeah next week will be high school math week fundamentals week and then the week after that will be number theory week.
So we got we got algebra uh yeah fundamentals number theory algebra rinse and repeat.
I know that of course there's a lot of crossover between number theory and algebra. But that's okay. There's enough number theory that you can study without algebra. Elementary number theory.
[clears throat] Why did Why did your internet die just when I was explaining the switch up?
Sorry about that. It's just a slight switch up. I mean, I'm just saying that I'm going to work like a few extra problems from these other books as well.
All right. You sure these will be enough for the whole week? Uh, I don't know.
Yeah, I might have to pull some more from back there. Yeah, there's a couple more that I could argue or algebra back there. We could we could pull some more materials if needed.
Okay, so let's put this to the side.
This monstrosity. It just makes me happy to look at this giant stack of books.
Um, Luna is going to knock this over and I'm going to be very sad.
Actually, I'd be impressed if she knocked this thing over.
Um, okay. Hey, my attendance, what's up, Rot Jr.? How's it going? All right, let's get back to this. Uh, and once again, episode 4 is out now in my basic math series.
Yeah, dude. I mean, I don't want to I don't want to, you know, I don't want to beg. I don't want to be like, come check out my my videos. I just want to make such a good math channel that Andrew York wants to show up, you know? Like, I just want to make such a because I have more ideas. I have ideas of how to just keep making the show better and it's just a matter of time and just doing one thing at a time and just keep getting better and better. Um, I want to make it so good that Andrew York wants to show up cuz like of course I would love Andrew York to be here. Are you kidding me? Like I'm a classical guitarist who loves math. Like and you know I can prove it. I've got a whole math series and everything. I'd be shocked if he hasn't been offered, you know, the the fundamentals basic math series yet. I'd be shocked. Um, but maybe I'm just like overplaying in my head how many people have been have seen or been offered that first video, but 3.2 million impressions is a lot of impressions. That's a lot.
That's a lot of people.
Like for sure YouTube is like offering that video to people who are not math people who are not already looking for math stuff. So I'd be I'd be surprised.
Uh oh, I I just saw this other comment.
Kathy Za, certified high school math teacher since 1988. Joining your live streams. Let's go. Dude, I'm trying to become a high school math teacher myself. So, any tips that you might have for me, I would love to uh uh to get from you. Kathy, never did go to graduate school for my enjoying math.
Thanks for your enthusiasm. Same. I mean, I've learned a lot of graduate level math, but for myself, and I took a lot of grad school classes in my undergrad, but then I never actually did a master's degree. Um, I love it. um you can still study that stuff, you know, even without a piece of paper like Yeah.
Okay.
All right.
Representation theory, finite groups. Uh and one last time I will say for anybody who wants to to see the episode, episode 4 is out now. Episode 4.mmathbahhobby.com.
Episode 4 out. Now look at that handy handy dandy URL. I bet you can guess what the URL would be for episode three or two or one or zero.
I I bet you can guess.
Um, okay.
Let's get started.
All it does is just redirect to the YouTube video. So it's not it's not anything fancy.
Um, okay. So I'm getting the iPad prepared.
Thank you everybody for tuning in and for hanging out with me. 10 likes equals 10 push-ups. Let me know when we get to 20. So um, I'll do 10 more. I'm not going to It's not like a triangle number. I'm not going to do like 10 and then 20 and then 30 and then 40. Um I would die if that happened. Um but still I might die even if we're doing 10 at a time. So hit the like button. Um and let's try to kill me with push-ups, dude. Let's do it. Uh representation theory chapter 3 group representations.
Um yeah, we got pretty far yesterday.
Okay, I feel very good about how far we got. We're almost to Mashki's theorem.
Yesterday we stated Mashki's theorem. Um 20 is live. Are you serious? Are we at 20 now already? Dang, dude. Okay. Well, thank you.
Dang.
Okay. Well, thank you everybody. Here's some of the math that we're about to hit. Um, wait, 30? Are we at 30 likes?
No, I see 21 likes. You a liar.
Somebody's a liar. All right. Well, thank you so much for 20 likes. Here's 10 more.
Hell yeah, dude.
What is Blackbird?
Yeah, I saw 21.
YouTube will be kind of funny about YouTube will be funny about um showing the number of likes. 26. How many?
Tell me when we get 30. Tell me when we get 30.
All right. So, we're all we're almost to Mashkis theorem.
What is Mashki's theorem? Mashki's theorem is similar, no pun intended, to uh factorization of numbers. Num every number can be factorized into a unique prime factorization, right? Like 30 is 2 * 3 * f wait no 30 is yeah yeah yeah 2 * 3 * 5 uh so 30 is a composite number um 30 is also 2 * 15 uh 2 is prime 15 is composite uh and then it's it can be completely reduced to a product of primes. Um, Mashki's theorem says that any representation of a finite group is similarly you can kind of factoriize it. It's completely reducible.
I think that that's the statement. Every representation of a finite group is completely reducible. Is that it?
Uh, yeah. Every representation of a finite group is completely reducible. I remember it. Heck yeah. Um and we need to So yesterday we stated all of these properties and we proved some things earlier. Um today I want to work these again. Uh not just stating them but also proving these properties. So first we have this property about um we have a property that a unitary representation is either irreducible or uh decomposible.
So irreducible is like a prime number.
Decomposible is similar to like a composite number.
um every representation of a finite group is equivalent to a unitary representation.
Um I think that this is now requiring that our vector space is over the complex numbers.
I think I think that we are doing a simplified version of this and I wonder if I wonder to what degree is that proposition true if we relax that requirement. So this book is an undergraduate an advanced uh undergraduate or master's level intro to representation theory.
um a a vector space can be a vector space with other scalers not just complex numbers. So I think that I think that this proof requires the fact that um that you can take coordinates of uh of a group element representation and and then we and then the complex numbers um have this inner product um structure.
Sure.
And we can show that the representation is equivalent to a unitary rep. Then a corollary, a correlary means like it's an easy result from a previous fact. A corollary is that um a representation of a nonzero finite group a nonzero rep of a finite group is either irreducible or decomposible.
So that is combining the previous two propositions because every unitary representation is irreducible or decomposible.
And so every rep of a finite group is either irreducible or decomposible.
Um and then finally this theorem by Mashki every representation of a finite group is completely reducible.
Okay.
I don't know what you all are talking about with this chemistry stuff. It sounds really interesting, but I'm I'm not going to I'm not going to get distracted by you all on that. Sounds really interesting, though.
Um, I've heard of Shimura varieties. That stuff is related to Fermas theorem, right? Shimmer Takana. Shimmerra.
Oh, wait. No. Shimmer.
Shimmerra is related to some theorem of some number theory thing I worked on in undergrad.
Uh huh.
Interesting. Okay, let's keep going.
So 3.2 point.
Let's rework 323 which is a unitary rep is either irreducible or decomposible.
Yeah. So we're starting from this point.
Um what's something heavy that I can use to keep this book open? Let's just use a couple of notebooks.
Okay. So, we have um proposition 3 2 3.
Formally it says let fee be a representation of a finite group.
No, let fee be a unitary, a unitary representation of a group.
Then fe is either irreducible or decomposible.
I was very close in my proof yesterday on this. Today I want to read and work the proof from the book a little more carefully. But I do remain I do remember that I do remember that um um I was extremely close on this one and I had the basic idea and I just had a slightly different argument but my argument was correct if I tweaked like one little thing. So let's let's read this proof. So suppose fee is not irreducible.
I remember I made this starting um approach as well. If it is irreducible then we're done. Um so if it's not irreducible, we want to show that it's decomposible instead.
Um then there is a non zero.
So if it's not irreducible, there's going to be a nonzero non-trivial um subspace of the vector space which is G invariant.
So here's my shortand for what this the rest of the sentence is going to say.
There's a nonzero proper G invariant subspace call it W of U. Ah, okay. Let's remember that when our vector space is unitary, we want to use the letter U instead of Wait.
Okay. Yeah. Yeah. Yeah.
Um I don't like I don't I don't like this right now that um hm be a unitary representation of a group.
Oh, there's kind of a weird uh mixture of notation right now in the book.
To me, this is a little confusing right now because previously we have a definition of a unitary representation.
A unitary rep is if um Fiji is unitary for all group elements.
In other words, um, VW equal the inner product of V and W is the inner product of the transformation of V and W. Um then we can consider fee as a um as a map from the group to the unitary linear transformations of V.
So we may view FE as a map from G to U of V.
So what I don't like right now is that U here is taking place of GL. So this is the set of unitary linear transformations which is a subset of all invertible transformations.
Um but then over here U is springing out of nowhere as a alternative label for the vector space. Right?
There's a non-zero proper G invariant subspace W of U.
like like I would ex like it it seems like as if they're saying that we can label that vector space as u gl of you.
Okay.
Rod says it's better to think of unitary as an extra structure on a representation the G invariant inner product rather than unitary rep as one noun and then say every rep is unitizable.
Okay, interesting. Every rep is unitarizable.
It's supposed to say W subspace of V.
Yeah, exactly. Like like in this sentence it should be V here. Like I wrote my proof this way. Like I was writing oh a second ago W is a proper subspace of V. And then I remembered oh yeah we can use U as a notation to imply that it's uh that this is a unitary map. So I was expecting that U is defined previously in the book as GL of U. But then I looked back and I saw that it's the book is using U as a subspace of the of GL.
And so there's like a weird mixture here, right? Where U is used in a completely different sense there versus here.
So I I feel that that is I I feel that this is uh a notation that he didn't update.
I I don't know. I feel that this was at one point when writing this book, this used to be GL of U, but then he wanted to use U to mean the set of unitary linear transformations or something, right?
Um, and yeah, this book Paul Getty, this book is uh Steinberg.
Um um it's like kind of arguably graduate level algebra part of graduate level algebra. Um, okay. So, that's where I feel [sighs and gasps] this proof is a little odd. It's not quite like I wanted to remind myself how do I use you in the in the shorthand for a unitary transformation and the book does it this way, which is a different way of using you. So, I think that this one is either a typo or or something.
He says a a proper G invariant subspace of U but it should be a V based on how this problem was stated or this uh theorem this proposition. Uh okay so that's just a little tangent. Um and then exactly as I did in my proof, um it's orthogonal.
It's orthogonal complement.
Um, WP is then also non zero and V is W direct sum with W per. This is a theorem of inner product spaces.
If you have an inner product space, any vector can be um what's the word? Broken into its orthogonal parts like the orthogonal you can find the orthogonal uh components of a vector that add to to any vector.
That's it. That's true in any um inner product space.
Um so it remains uh to see to prove that WP is G invariant. So this is exactly how I did it. So it remains to prove that WP is G invariant.
If V is in WP, I don't like that not that choice of a label, but we'll live with it. If V is in WP and W is in W, then the inner product of G of FGV, I'm going to emit some parentheses. I think it makes things look better.
uh we want to see that this is ultimately zero. If so then um then the image of that vector is in WP.
So G WP or WP is G invariant.
This is what we want to show and we're going to get there eventually.
Okay.
Um, I remember that the book had a really kind of cool argument about this one. I remember it for myself.
For any group element, there's an inverse element.
And so the by the fact that fee is unitary then this is true.
Uh so then because f is a homorphism it's f of the identity element.
Um because w is g invariant. This is some new some new element in W and the image of the identity. Always the image of the identity is the identity in the target space.
Um, and so it's the identity map of V, which maybe I could write as E identity times the vector is the vector itself.
Uh and because V is in W, V is orthogonal to everything in W. So this inner product is zero.
And then we're done. I'm pretty sure that this is how the book does it. Let me confirm and let me just uh see exactly the steps that he takes. He doesn't do as many steps as I did.
um doesn't fill in as many details, but it's still the basic the exact same basic idea that this is FG inverse FG VG inverse W and then Yeah, Fiji inverse is going to undo Fiji. So that so V is fixed and then we have F G inverse W and he says that this is zero.
um he gives these labels and then a description.
So this is 3.1, 3.2 and 3.3 and then explains the steps afterward and says where 3.1 follows because fee is unitary.
3.2 two. That next equality follows because um yeah, FG inverse FG V equals F of 1 uses one. I I've seen E more often, but either way, the identity element and then that is the identity transformation.
I hesitated to write capital I because this is a transformation, not not a matrix, even though you can kind of use a correspondence on those, but okay. He used capital I. That works for me. I'll I'll get used to that. Um and 3.3 follows because yep as as I see uh because f g inverse uh w is in capital w as W is G invariant.
Yep. Yep. Yep.
And V is in.
Okay. So we I did an extra line just to give Fiji inverse W like a W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W prime or something, but he's doing both in one step that Fij inverse W like W is um G invariant and V is in the orthogonal complement.
So then the inner product is zero. Yep.
Yep. Yep. We conclude we conclude that W per is oh wait no we conclude that fee is decomposable.
Yeah.
And then you get to draw the cute little square. We are done because we already said earlier that our goal, our only goal that we need to complete is the fact that uh WP is G invariant. Yep.
[laughter] Yeah. What's up, dumb Peppa?
Yes, exactly. Yeah, the proof today is actually correct. Uh, yesterday I got like the basic idea, but I needed to to fix it fix it up a bit. And I remember I was like, yeah, I need to take an arbitrary vector in in one uh space and then show that the inner product is zero. So yeah, I just wrote out the exact argument from the the exact proof from the book and but I I still saw like this is the exact same thing and I I I was just reviewing what yesterday's work was yesterday. When did this channel get so big? Um uh Dumb Peppa, have you seen this video, bro? Are you still there, dude?
190k plus it's nearly 200k views now.
Um Dude, how do I Oh, my books are in the way. Here it is.
Yeah.
Yeah, that one video is has nearly 200,000 views and it's just an announcement video. That's kind of amazing. And a lot of people from that video are saying, "How do I pay you? How do I sign up for the for this course?"
Like that's kind of kind of incredible that like my video announcing a course or a series, an introduction video.
I I feel that YouTube's going to be recommending this video for a while. I don't think it's going to be hitting a limit. I think it's slowing down. It's tapering, but it's going to continue growing.
You know, we're I feel it's going to grow more like the harmonic series than the, you know, reciprocal squares. The sum of reciprocal squares.
Just got to grind more and more until you start to understand what to apply when. Yeah, JC, apparently your idea definitely fills a gap here on YouTube.
Thanks, man. I think so. And I mean, that's what my video is focused on. I was like, I think this is under represented. There's a lot of people who study math as a hobby and we don't talk about that enough. And yeah, that's that's turning out to be true. I think that that is the new idea. There's a lot of videos out there sharing math for fun, you know, because math is beautiful, but I don't think anybody I don't know of other big videos that really focus on the idea that it can be a hobby. Like you can study at as an adult just for fun as a hobby.
>> Yeah.
Yes, Jonathan Ward. Thank you, dude. Um, >> yeah, we are going to do an averaging trick in the next proof in a moment. Um, and yes, this book does assume that the whole book is over complex scalers. I know that that might be a little underwhelming for some people, but I think for an intro to um you know, representation theory, I think that this is nice right now. But I I do want to get other books. Which books would you all recommend? Does how work over complex numbers or does it also does how howie also um simplify things with complex scalers or or yeah curious I I bet um Harris and Fulton I feel like y'all are talking about like Harrison Fulton or other standard treatments like sir or Harrison Fulton Right.
There should be some kind of disclaimer somewhere. Yeah. Like note or warning.
There should be. Yeah. V is over the complex numbers.
Warning. Yeah, people do ask that a lot and it's because you all know this topic more than I do. I wouldn't even know that this is an issue because I'm I'm you know learning this uh for the first time and I forgot that yeah the we're working over complex scalers.
That's cool.
Oh, dominant foot does it over a general field. Nice. Awesome. Awesome.
Yeah, I'm man, I'm having fun with this and like you all are helping me a lot by, you know, filling in some of these details and if I get stuck on a proof or something or I write a proof that's incomplete, you all are there to help me out. So, this is like just such a cool way for me to be learning and practicing this. So yeah, Harrison Fulton is too much of a classic to not suggest even beyond representation theory. I will get it at some point. Yeah, we will add in Harrison Fulton to this stack. Um, for anybody who just tuned in and wasn't here for this earlier, I am now announcing that my stream structure is changing slightly.
This week is representation theory week as usual, but it's also more broadly algebra week.
This week is algebra week. Um, I'm not going to pick up the stack, but there's like half of the stack right now.
Everything below dominant foot is on starts the top half of the stack and then there's another half below it. Um, so this week is algebra week and uh basic math doesn't belong in there, but still. Yeah. Yeah. Let's let's take basic math out of this stack.
That's kind of a stretch. I feel basic math is a stretch in this in this stack.
This week is algebra week.
I got a bunch of GWA theory books. I got dominant foot. I got Aluffy in there.
I've got Eisenbud at the bottom. We got Big Boy Lang up top. Um, I don't have Harrison Fulton yet, but I'll get it at some point. And my point is that I will be primarily focusing on Steinberg.
Yeah, I will be primar primarily focusing on Steinberg, but um I do want to add in some exercises from easier books like Gallion and potentially even harder books as well.
[clears throat] Excuse me. Okay, so this proof is now a bit better. We do love algebra, don't we? Don't we? I could just keep that up. You don't need to see my face, right? Like, why? I just want like one or two more books and then we would have a beautiful stack.
[snorts] Um, I've got some number theory books, but we're not putting number theory in the stack because number theory is its own separate week. So, a week of fundamentals, a week of number theory, and a week of algebra. That's the new stream structure.
[snorts] [snorts] [snorts] Um >> [snorts] [snorts] [snorts] [snorts] >> Okay.
Well, I my parents just shipped and sent like a bunch of books that I have from college. Um, I don't see another book that I truly with a straight face can call algebra back there.
Um, [snorts] not number theory.
Thank you. Thank you for blessing me. Um, [clears throat] but I do have string. So if we're calling linear algebra part of the algebra series, we do have string. And I think that this completes the stack.
[snorts] Does it take up the entire camera frame?
I could just cheat and crop the very top. We're so close. You can still see my hair just a tiny bit. Random question chat. There are only two irreducible polomials of degree 3 and f2x right. Uh I don't know what you're talking about.
I mean I kind of do but thanks Yun. Um y'all we just need one more like. Now's a perfect opportunity before I start the next little practice session. The next little practice thing. Um, [snorts] yeah. We just need one more like and I'll do 10 push-ups and we'll start the next. Thank you so much for the 30th like.
Appreciate it.
All right. Hell yeah, dude.
[snorts] I'm going to go blow my nose. Be right back. Super quick. And uh and then we're going to start the next exercise or the next proof.
Okay.
What's up, chat?
[snorts] Cool. Let's keep going.
Next one. Do should I just keep this stack up? I mean, I like the way it looks personally.
Did OBS crash? Oh my god. Okay. I thought OBS crash.
I mean, I like the way it looks.
You don't need to see my face, do you?
[snorts] Um, all right. Let's get started on the next exercise or not exercise but the next um proof.
[snorts] I feel ready to call this one done.
Going to cross it off.
Uh prop 324. Now this one it was a really pretty argument I remember. Uh so it turns out that for finite groups every representation is equivalent to a unitary one. This is not true for infinite groups as we shall see momentarily.
Whatever makes you feel comfortable. I don't care either way. I just think that this is a nice way to uh announce our new Okay, let me reframe.
This week is algebra week. Next week is foundations and then the week after that is number theory.
Foundations, number theory, algebra.
Foundations, number theory, algebra.
Foundations is primarily basic math.
But yes, for those who know, if you know, you know. We're going to do other foundations as well in foundations week.
Okay. Proposition 324.
If you know, you know.
Every representation of a finite group.
G is equivalent to a unitary representation.
Okay, this one [snorts] I'm going to start out a little bit on my own, but I know that I'm not going to make I'm not going to get that far into it.
Um, [snorts] I know that for this one I'm I'm going to have to read the proof, [snorts] but I know that um, let's just keep saying this, warning, we're working with vector spaces over complex scalers.
We assume throughout this book that vector spaces are over the complex numbers.
The vector space itself doesn't need to be complex numbers, but would be better if the books magically turn into a cute cat. Well, that happens. Uh, Yan, do you or Truisatia, do you know that that happens all the time? Actually, if I put a book, like one or two books here in this place of the desk, Luna sits on them sometimes and she will just photobomb and just be super cute. I hope that that happens today. We'll see.
I guarantee that will happen.
Um, okay. Okay. Okay. Okay.
Every rep of a finite group is equivalent to a unitary rep.
So we want to find another representation.
um to another vector space such that FE FE is similar to our second representation.
In other words, we want an isomorphism t um such that this diagram commutes.
So either path either going this way or going this way.
Um you have the same the same result with taking either path around this uh diagram.
So that's saying that taking s after t for a vector in v1 is the same as taking t after v.
Um, I mean I could use V and W. I I wonder if V and W is is nicer.
Uh so in other words [clears throat] S is T F T inverse for any group element G and I think because we're choosing um V over the complex numbers which is this big warning that this book is limited to um vector spaces over complex scalers.
Let t be the map from our vector space V [snorts] to Um do complex vectors complex valued vectors given by the map V to taking coordinates. That's [snorts] [snorts] so we've shown already previously in the book that um that this map is an isomorphism. In fact, I think Hey, hey, Talascar.
Good to nice to meet you, bro. Um, thanks for asking.
I will give you a moment here to answer that question. Turge, you asked, "What is my setup?"
You want to start streaming? You should start streaming. It's awesome. It's just super fun. Um, I use OBS on a Mac.
I use an iPad with the Apple Pencil and uh the app is Notability.
[snorts] Um, and then I use an app on Mac called Airerver um to wirelessly display my iPad on screen.
Um, I think that you can figure it all out with this, but if anybody wants me to help them set up everything and just like do it for me, like I can I'll I'll help you out with uh for like 20 bucks uh per hour. Um, I could help you set this up over a Zoom call.
Um, especially OBS has a million. It's like a steep learning curve and I promise you that would be 20 bucks well spent, dude. Like OBS takes a while to learn and I wish I knew about certain tricks about it early on.
Okay. So, I I think that we got the picture about um algebra week. We're now in algebra week right now. Um I can put my face back up.
All right.
[snorts] I think you should keep your setup info in the description. Yeah, maybe.
Maybe.
Jonathan Ward. I need to understand that better. How will Luda photobomb this setup? She, if she photobombed this, she would knock over the stack and it would go tumbling down and it would be a delightful moment.
[snorts] That's how she would photobomb this stack.
Um, [snorts] okay.
I can help you set up everything.
There's a lot involved. Um, thank you for noticing and appreciating it because there's Yeah, there's a lot of stuff involved in this and I'm proud of it.
It's taken years for me to kind of land on how I like to set up stream. Also another important one is um stream deck is really important in my entire setup.
The Elgato stream deck is really a pretty pretty big key in keeping everything reasonable.
Um okay.
All right. But I I think you can figure it all out from that. Those are all of the key ingredients.
[snorts] Um, okay. So, I I can take coordinates.
[snorts] Um, [snorts] it remains to be seen.
We need to define um GL of C to the N is isomeorphic to GLN C.
We need to define sigh [snorts] like this. So again we use this equivalence everywhere.
The glnc is the same as gl of cn.
Um, we need to find a unitary representation which we have some hope of doing because CN is uh an inner product space.
>> [snorts] >> But it doesn't need to be the standard inner product.
We define a new inner product.
as how does this go for two vectors um V and W the inner product of V and W I remember is this pretty cool averaging trick which apparently is useful everywhere.
Hey, hey, SOG, thank you, dude. Thank you for the domain of Sai is looking a little funky. Is it? Oh, my bad. Thank you.
You're right. Yeah, yeah, yeah, yeah.
Appreciate that. That was a typo.
Good call.
>> [snorts] >> Um, bro, thank you so much, SOG. I I appreciate that very much, dude. Um, put you in the thanks page here, SAG7611.
Appreciate you, bro.
Um, we are Let me also give you some credit here.
Thank you very much, dude, for supporting my journey, bro. SAG 7611.
Look at this.
Look at this, dude.
Uh, you're down there along with SR71, Levi, James Wahberg, and SAG. Thank you all for supporting my stream and my journey.
I really appreciate it.
Um, okay.
So we have this averaging trick that I remember that is going to be defined as the inner product of um how is >> [snorts] >> Is it just this?
Yeah, we average over all of the group elements.
Because SIG GV is a [snorts] n dimensional complex vector or a matrix.
I think it's this. I think that this is right.
And then I read this privately. I'm not coming up with this right now on my own. That would be amazing. I think I I cannot pretend that I am inventing this. And um recall that sigh is I have to double check this all the time. T V T inverse T F V T inverse.
So this is T F V F T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T inverse.
Oh, wait.
[snorts] Do I already know?
I mean, yeah, I know that we have uh [snorts] H.
I know I have this isomorphism t but I haven't yet defined sigh right I oh no no no s I can say is defined as this I can I can define s as this because f exists t exists t is taking coordinates as a as a map which is an isomeorphism. So we have the inverse as well. So sigh is well defined as this thing and now we just need to show that sigh is itself unitary.
Um with respect to this inner product so um I think that this is what we define as We defined this as an inner product and we need to show that this is an inner product.
And then secondly that this is unitary i.e E S V S W equals VW for all vectors V and W, right? Hey, what's up int? Yeah, you're um Divia, right? How's it going Divia?
Today in your test you did some multiplications.
Nice.
How's pac calculus? I haven't worked it.
You mean sigh is unitary for I got that right. I fixed that.
Oh um I see. Yes. Yeah. Yeah. Sigh is unitary.
You're right.
S is unitary for this inner product.
Thank you.
So S is unitary with respect to this inner product.
Um okay.
For one, we check that um if you flip like we have this conjugate symmetry.
This is one of the axioms.
What else do we have?
We have a like a linearity thing.
And we have this um positive definite something or other.
Is there anything else I'm missing?
Um so let me try.
So right now our goal is that this defines an inner product first.
Um, WV is defined as This and where that is the standard inner product on complex um vectors and this standard inner product has that conjugate symmetry. three.
Um, and the sum of conjugates is equal to the conjugate of sums.
I think that's always true.
I should be a little more confident about that. But I'm pretty pretty sure that's true for any two complex vectors.
Uh oh wait, I wrote this more in this direction.
If you sum two conjugate vectors, it's the same as the conjugate of the sum.
Uh and now this is the conjugate of vw as desired.
So this property works.
We got that one.
Secondly, um I think that this one holds by linearity and we're we're taking an inner product across all group elements.
So because s is a linear transformation itself. This is k si v si g w over all group elements.
Um because this is the standard inner product.
This is K Si V Si W.
Um am I missing I am missing um linear addition aren't I like the book actually writes this as one step of linearity but the other book I use does it a little different.
We want this to be UW plus VW, right?
Bil linearity is a thing. Yeah. Yeah. We need both of these and you can write it as as one. And as long as you combine that with this this first part this first property then I don't need to write linearity in the second slot right there's conjugate linearity in the other argument that's what I'm saying right now I think I mean if you just combine that with the first one then you automatically get that conjugate linearity Right.
Um okay.
So this is averaged over all group elements.
And now uh I can factor out that K out of every term.
This is just for funsies at this point.
I already see that this is working out.
All right. So now we have um linearity of a scalar multiple.
linearity of addition.
These basically hold by li by the properties of the standard inner product and summation.
I [snorts] don't know how to put that.
This is defined as um ci oh wait no fe uh we use the fact that s is a linear transformation. information.
So we get s g u plus s g w si g. Oh wait no side u v and w uh as we expect. Now we get now we use linearity of the standard inner product.
Ran out of space here.
Scooch that over just a tiny bit.
Uh okay. So now we [sighs] use linearity of the standard inner product and now uh we can split this as two summations.
And now we have the inner product of like our our funny smoothed inner product of uw plus vw.
Okay.
Um, what do I call this? Positive definitess.
Yeah, that implies conjugate linearity in the other argument. I think I I'm pretty sure. Yeah, that should um cool.
Uh finally to show that um that this smoothed inner product is an inner product take the inner product of any vector vv as the average inner product >> [snorts] >> Um, this is greater than or equal to zero.
Okay, we'll see.
Let's rework this.
This is greater than or equal to zero because it is the summation of positive of non- negative values.
What's a good way to say summation of non- negative values?
So finite sum of positive values with x greater than or equal to zero with every term every term is non- negative. I I think that's a better way to say this because every term in the sum is non negative.
Furthermore, VV equals zero if and only if every term um G VI G V equals zero because every term is non- negative.
negative. The only way that the smoothed inner product is zero is if every individual term is zero, which is true by the standard inner product if and only if FGV equals FG V.
Oh well, no. um which is true if and only if fguv is zero and because sorry I I keep saying fee I meant sigh this is true because s is an invertible linear transformation. This is true if and only if v is zero because s is an invertible transformation of the vector space into itself. Right?
Uh that splitting of summation is doing a lot of heavy lifting with g being finite.
Yeah.
Yeah. Yeah. I think this splitting of the summation this is needed that g is a finite sum. So we're allowed to to split that summation. So technically you only need the size injective for the last implification.
Yeah, good good call. Good call. So here, thank you dude. Um note we are really using the fact that um that the summation is finite. So I can split it into two summations.
Technically technically we only need that it's injective. Thank you.
Um so therefore this defines inner inner product this smoothed inner product.
Take S G V S G W and we want to show that this is equal to V W.
This is our goal.
This is defined as Oh yeah, I think I remember how this goes. It's super cool.
Okay, here there's an overloading of the term G. So we want we're doing this over all group elements whereas G is fixed to the group element in um yeah G is is one fixed group element in the group capital G. So then the summation needs to be over a different index.
Um Because sigh is a homorphism.
CI G SIH equals S of GH.
And now we claim that because G is a group, multiplying every element in the group by G fixes.
How do I Right? Like if if a group is finite then every if you multiply every element of a group by a single element you you still get the same group right like for any group element G lowerase G times capital G is equal to capital Right, sir. That should be true.
I feel like I should be able to prove this really quick, but this seems like it's true. I kind of think of extra assumptions or as unused arguments and Ziggs LSP has made me really pay attention to unused stuff.
Yeah, you need the summation variable to be different from the G outside of scope. Yep. Yep. Variable name shadowing. Yep. programming parallels very much. You should swap the order of H and G technically. Do I need to Oh, you're absolutely right. Thank you.
You're absolutely right.
Yep. Yep. Yep. This should be um site G here and site H there. You're absolutely right.
Why am I feeling uneasy though about what's the name of this that I'm claiming? If you multiply every element of a group by any single element in the group, you get the entire group again.
It doesn't fix H, but it does get every element. You can see why. Oh, yeah, that too. It's a simple proof that you can try right now. I know. I just I feel like this should be um if G is a finite group then multiplying every if you fix a single group element.
Then um for any for any element in G um H is G * GI for um group element um so I take H n then Um um H is the same as H g inverse. So I know that G inverse exists.
So H is the same as H E where E is the identity.
E can be re can be written as H G times G inverse which I know exists because every element has an inverse a multiplicative yeah an inverse element.
Uh, and now, oh, maybe I want, uh, yeah, I just want this slightly different to to match exactly the order that I wanted. So, H is the same as E H.
Uh, E is GG inverse H. And then now whatever G inverse H is this is some this is some element GI in the in the group list.
So for all for any group element in G then for any other group element in in G I can express the second element as G times some group element that some group element being G inverse H. Yeah, I think I got it right. Every element exactly once.
That's Yeah, I'm trying to formalize that like it's it like cycles the group.
It's it's uh what's it's an it's a bjection.
Multiplying but by any group element of the group is a bjection.
Yeah. Oh, there it is. I multiplying by an element defines a bjection on the group. That's what I thought. Yeah.
Oh, yeah. Yeah, you're absolutely right.
Oh, no. I said GG equals G. So equals here I think is is implying byjection. I didn't write like subset.
Yeah.
So I think that this is it, right?
You can solve for GI from just that equation.
I think I got this.
Hey, what's up? A Pong Dynasty.
You got the subjective part. The injective part should also be easy to see.
Okay. Um, yes, this is on two. And then um injective says that if what would we be saying? We'd be saying that G Am are you all hearing audio like computer audio? Somebody would have said something if so. Okay, never mind. Um, so if GH1 is equal to GH2, then H1 equals H2. This is what we're trying to say. Um and then that's clear by multiplying both sides by G inverse which we know exists. So multiplication by a group element defines a bjection on the group. Nice.
Yeah, you're not you're not hearing background computer audio. Good. Okay.
So this now defines a bjection on the group. So this step is now justified that for this finite group we um every element of the group is repres is is listed like HG equals S I and this summation still holds right. Um and now this is again Um now this is the inner product the smooth inner product of v and w. So therefore um s is unitary right.
Heck yeah.
Okay, so I I did read this one a week ago in the in the library or well at a cafe.
Uh so I'm glad that I remembered it. I don't think I missed anything important.
I think I got everything for this proof.
Um because yeah we defined the we defined the isomorphism t being um taking coordinates. So we we defined what s is in terms of [snorts] that isomorphism with uh with f.
So sigh is well well defined and then we showed that sigh that we can build an inner product for which s is unitary and then therefore we have found an equivalent unitary representation of any representation.
I think we're done.
Let me read the proof. Let me see.
Okay.
So they do assume that the dimension of the vector space is n.
Okay. So do we need to assume that the dimension of V is N um in order to define T from V to CN by taking coordinates.
Yeah, I think we need to assume a finite dimensional vector space in order to take coordinates.
Yeah, choose a basis B for V and let T from V to C and D the isomeorphism taking coordinates with respect to a basis.
Yeah. So this isomorphism only exists if V is finite dimensional which I think that the I think that the whole book is generally assuming finite dimensional vector spaces but here it is really clarifying like we need to assume a finite dimensional vector space here so that we can make that map well defined.
So that part was missing.
Usually it is usually the book is stating that these are finite dimensional finite representations like finite dimensional vector spaces but not always.
Um okay so choose a basis let t be the isomeorphism taking coordinates with respect to that basis then setting they use row for some reason um let row g be exactly as I defined s uh this crucial averaging trick will be a frequent player throughout the text.
Yeah, finite dimensionality is assumed through throughout the whole book. I I think so. I don't see right now offhand where they state that we assume throughout the book, but like the book does say specifically every time that um that we are assuming like when we use the fact that it's a finite dimensional like I'm sure it says it somewhere in here but I'm just saying like I forgot to to stipulate that yeah we do use the fact that it's a finite dimensional vector vector space like to define this isomorphism to define taking coordinates and choosing a basis we have to to say that dimension of v is n.
Yeah.
So it's not stated in the statement of the proposition because it's already assumed that V is finite dimensional but in the proof we use [snorts] finite dimensionality.
Yeah. So it's just like a subtle thing that I missed but that is needed I think. Uh is my iPad dead? My iPad is dead. Intro of chapter 21. Yeah, I figure it's it's in there, but we use that fact in this proof and I just forgot to to mention it. I am in my head always assuming complex scalers and I'm always assuming finite dimensional vector spaces, but I just I forgot to to use that fact in this. Okay, cool.
um summation over g of course requires that g is finite. It can be viewed as a smoothing process. Let us check that this is indeed an inner product.
Um we check so they check both linearity of addition and multiplication at the same time and then they check um conjugate symmetry.
And then they make the same argument that um the inner product of a vector with itself is greater than or equal to zero because every term is greater than or equal to zero. If it is equal to zero then every term must be zero. Um, so why do we uh let's see page 22. Top of page 22 I'm a little unclear on. Um, okay. So if VV is zero then we get that which implies that for all G. Oh okay.
So in particular oh wait oh yeah inner product of FijiV Fij is zero for all G and G. So in particular we can see that for the group element of the identity element the um yeah I just did something slightly different to show that v equals zero but that is a little nicer I think like what he wrote. uh to verify that this representation is unitary with respect to this inner product, we compute this and we now apply a change of variables by setting x is gh as as g ranges over all g x ranges over all elements of g.
Uh since if K is in G then when D. Yeah.
So they made the same type of argument that it cycles the entire group. Okay.
And then that completes the proof. We got it. We got it.
Nice.
Your favorite top three mathematicians?
I don't know. Uh, Galwa, Gaus, Galwa, Gaus, and um I don't know.
Girdle.
I think Galwa, Gaus, and Girdle would be my favorite. Oiler is up there, too. But I I was thinking Oiler or Girdle. I think Girdle beats out Oiler for me.
No. No. Growth and Deer. Oiler. Yeah.
I just love Girdle's theorems, man. Like it's just so cool and in a different way.
I I could be wrong about Oiler above Gaus for me.
Um, I don't know. How do you pick? How do you pick?
Um, okay.
Is my iPad What? What happened to my screen? What are you all seeing? Yo, we just need one more like and I have to do more push-ups. Come on, somebody hit the like button. We're so close. I I should be doing more push-ups.
Um, is my iPad alive again yet?
Should probably go soon. We're two and a half hours in.
Oo, on the finite dimensionality point, it might be a fun exercise at some point to show that if V is not finite dimensional, then it has a finite dimensional G invariant subspace.
That sounds cool. I don't know anything about infinite dimensional vector spaces.
I don't know anything about it.
Sounds cool.
Sounds really fun.
Okay, I want to I want to get to the end. So now I am just going to read and copy the proofs of the next two.
I feel pretty good that we got prop 324.
I remembered it. I was able to do it from memory.
I got pretty much all of it from memory.
Um, corollary 325.
This is a short one. Oh, I think I correlary 2 325. I saw how that's just like immediate from the others.
So, yeah. I I saw last time how the corollary holds. We have an example that I'll understand later. And then we get Mashki We may need this corary.
um okay so if f is a nonzero representation of a finite group it is either irreducible or decomposible um and that's because v is equivalent to a unitary rep And every unitary rep is either irreducible or decomposible.
Right?
So why do we need uh because it's equivalent to a unitary rep and then because it's equivalent to one and then the other lemas which said that if a rep is equivalent to an irreducible or equivalent to um decomposible then it is itself irreducible or decomposible. So yeah we you also need the lemas that say equivalent to irreducible implies irreducible and equivalent to decomposible implies decomposible. Okay, I got that. Now, Mashki, every representation of a finite group is completely reducible.
Heck yeah.
Um okay so so far we have every representation of a finite group is either irreducible or decomposible, right? And [snorts] if v is irreducible, you're done. If FE is decomposable then that says that if Ve is decomposable then you can consider it as um a direct sum of two representations which themselves are um which themselves are representations of the finite group and every step along the way you're reducing the size of the vector space.
So at some point breaking down like every every composition of the vector space is reducing the dimension of the vector space and I think because V is finite dimensional this process must stop at some point. So you hit um a factorization into irreducibles.
So then once you have a factoriization of irreducibles then it's completely reducible.
Yeah, I think I have the idea of this thing. Let's write it out formally.
Dang it. Come on. Delete.
So, let fee be a rep.
of a finite group G.
The proof proceeds by induction on the degree of FE that is the dimension of V.
I see the proof proceeds.
Yeah.
By induction on the degree of FE i.e. the dimension of the vector space V.
Um if dim V is if we have a one-dimensional vector space then Then the previous corollary says that um fee is must be irreducible. It can't be decomposed um into non-trivial subspaces.
Yeah. FE must be irreducible.
If dimension of V is one, then FE is irreducible. Yep. Then FE is irreducible.
Um since V has no proper Yeah. proper subspaces.
Yep. Yep. Yep.
Okay. Okay, that's what I thought. Um, so the base case is satisfied. Then we assume the theorem for an N- dimensional vector space. So I I think it's going to say something like if dimension of V is N And um and the statement is true for n -1.
for any vector space.
I think we only need weak induction.
Um maybe we need strong induction.
Then if fe is not irreducible then it is decomposible by the previous statement by the yeah by the previous corlary fee is either irreducible or decomposible. If it's irreducible we're done. If it's decomposible, you decompose it into um two proper subspaces and and we assume that uh that the statement is true for all dimensions less than n. So then it is the direct sum of two um v is the direct sum of two proper subspaces each of which are completely reducible. So then the parent space is completely reducible.
Something like that.
Uh has no proper subspaces.
Assume the statement is true for dimension of v less than or equal to n.
Yeah. So I thought we do need strong induction once I thought about it a little bit more. Yeah. So assume the statement is true for um dimension of v less than or equal to n.
Let v be a representation of a finite group.
Um, be a rep with dimension v= n +1. Yep. Um, I don't like the overloading of V in those in these two statements right next to each other cuz those are not the V same V. But that's okay.
Um, if FE is irreducible then we are done.
Yep. Then we are done.
Otherwise fee is decomposable by the corollary as I said otherwise fee is decomposible by the corollary.
Uh and so V is um V1 + direct sum V2 with where um both these are non zero G invariant. and subspaces.
Yep.
Since the dimension of v_sub_1 and the dimension of v2 are both strictly less than the dimension of v by induction.
Yeah, I I shouldn't have thought like specifically the restriction the subrepresentations by induction.
FE uh restricted to V1 and FE restricted to V2 are completely reducible.
Therefore, V1 is the direct sum of U.
I don't like U here because U is not unitary, right? Why not use W in this spot? Oh, okay. Yeah. Yeah, he's going to use W in a second. So U1 direct sum uh to us and um V2 is W1 is the direct sum of uh WR. Yeah. WR where the um UI and WJ are G invariant subspaces.
Those are G invariant.
and the subrepresentations and the subreps fee restricted to any of those um atomic um subspaces.
Oh wait, no. Then V is U1 direct sum.
This is just kind of a little tedious like writing out all of it, but um direct sum W1 direct sum and hence fee is completely irreducible.
Wait. And hence fee is completely irreducible, completely reducible.
There's a typo.
At the end of that proof of theorem 328, the book says, "Hence, FE is completely irreducible."
But I think this should say completely reducible even though that's they're kind of similar but shouldn't shouldn't it be completely I mean the statement of the pro of the theorem is completely reducible completely irreducible is yeah that's just a typo that's just a straight up typo neat Okay, we did it. We proved Mashki's theorem and we got to the last theorem of this book. Maybe tomorrow I will work the examples.
Try to understand a little more carefully. Um or like a little bit more maybe I'll work an exercise or two tomorrow.
Huh. Homotopi type theory intensifies.
Uh, 38 likes. Two more to go for another round of push-ups. Have I done the 40th?
Hey, still going. Yeah, I'm I'm about done.
I wish I could be more irreducible than irreducible.
Yeah, dude. We got it. We just finished chapter three in terms of understanding all the proofs of all the theorems. We got there, dude. I feel good. I feel like I just climbed a a small mountain, you know, like I'm like getting back in shape. I I used to I used to understand, you know, abstract math and we're getting back to it. I feel this is this is an accomplishment for me. Yeah, like for me at my stage of life having not gone down the grad school path like I feel good getting to the end of this chapter and you know I I think I'll I'll get an exercise or two in the coming days and soon I will uh have K theory on as a guest. I hope soon. I really like we've talked about it. We've talked about having K K on on stream as a guest and I want to try to lecture to him and explain my knowledge, everything I've learned over the last couple of months. Um, thanks everyone for watching. It looks like we just hit 40 likes. Let's do 10 more and then we're done.
All right, let's go.
Um, yeah, sleep well, Lock. Good to see you, dude. Good to see you.
Yeah, dude. Yeah, we did it. There's [snorts] 10 more.
Feel like they're getting better.
Take care. Thank you for watching. Um, please do like and subscribe if you haven't already. And dude, it would help me out a bunch if y'all want to check out this brand new video, new video, episode 4 of my basic maths series, intro to proofs slashmath as a hobby.
Um, here's a handy dandy shortcut to get there. Episode 4.Math can be ahobby.com.
New video. If you all haven't watched it, um, I hope you check it out sometime soon. And feedback is such a gift. Let me know uh what you think of how that video went on like the presentation style. Like if you know this math in and out, you know, how did I do as a teacher? Like you know, would you have explained like was there a point where my explanations are unclear that I should have moved faster or slower? Give more examples, less examples. Um, yeah, I'd love to I'd love to talk kind of math pedagogy even with those with those videos.
So, I hope you check it out. Thank you for watching. Take care everybody. Bye.
I had fun today. Bye. [snorts]
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