Day 128 – Practicing Math Live – Ch. 3 Group Representations

[00:00:00]Hey, what's going on YouTube? How's it going? I believe yeah, now we're now we're live on Twitch as well. How's it going everybody? Welcome back to my show. Uh my name is John. We will be practicing mathematics today. In particular, we will be studying some representation theory of finite groups.

[00:00:22]So, I work in week-long sprints. Uh this is an idea that I've taken from my time working in the software industry. So for one week at a time we're working on high school math. For one week we're working on college math. And now this week we're going to be working on graduate level math. And so for anybody who's watching these videos uh for the you know a newcomer people who are joining the channel for the basic mathematics series welcome. I hope you enjoy yourselves.

[00:00:56]This is not the basic math series. I just want to emphasize that. Um, but I hope you enjoy yourselves nonetheless.

[00:01:04]Uh, there's a good chance that most people hanging out in the chat, most people studying math in this graduate level series where we're working representation theory, most people don't know this math. I'm learning this math for the first time for myself. And I mean, I I live it. I claim that math can be a hobby. I'm actually living that.

[00:01:29]I'm studying this stuff for fun. Um, most people who study this are trying to become a mathematician. I am just I just like it. I just think it's cool. It's a It's a fun intellectual exercise to try to understand advanced math. Um, you know, so that's what we're going to be doing today. Uh, if time allows, I might film the next episode in basic math. Um, at the end of this stream live, we might film the next episode live. Yo, what's up, Success Road? El Cookie, good to see you all. Good to see you all. So, I'm still setting up the stream. Um, so Success Road, you found me because of the basic math series. How are you liking it so far? Um, are you enjoying the are you enjoying the episodes? I'm joined by Luna. She's over here on the floor. I hope that she joins us on the on the desk, but we'll see. Um, yeah.

[00:02:27]Yeah. Hope you all are enjoying. So, there are there's episode zero, the one that really, you know, became very successful. Uh, and then there's three episodes so far in the basic math series. So, I'm curious if uh have you seen those? Have you watched episode three? y'all. Um, I was liking and running away. Kind of busy with my own stuff. Okay. Oh, El Cookie. Yeah. Yeah.

[00:02:52]Yeah. It's been a little while. You haven't tuned into the streams most recently. I remember the series is worth it. Heck yeah. Heck yeah. That's I hope it's worth it.

[00:03:02]It's worth it for me. Absolutely. Um, I'm I'm very excited about it. I mean, I'm having a blast with that series. I think it's I think it's a pretty cool idea. Um, yeah, Louie or Euie, funny enough that you asked, we just we just ran the farewell tour to calculus actually last week. Um, there are I don't know at least 20, if not more streams that I've filmed um practicing calculus live. So, there's there's a bunch. You can find the playlist on my channel. Um, right now it looks like we're moving on from calculus. Uh, yeah, and we're going to get back to group theory. Group theory, groups and representations. For those who [cough] are curious and you haven't heard of this before, group theory is the study of symmetry.

[00:04:00]It's study of um it's they're mathematical objects that describe symmetries of of abstract weird objects. Um a simple one though I can show you with a well if it's if it's really folded then it it it's hard to do but like imagine a Rubik's cube. a Rubik's cube that you can spin around.

[00:04:28]It's and then it's a new cube, but all the vertices have been have been shifted. Um, it's kind of like this square.

[00:04:52]Okay. So, this square I wish I had uh That's not going to focus. That's not going to focus. One, two, three, four.

[00:05:02]Right. I can rotate it 90° and then I get another square, but all the vertices have moved. I can rotate it again. I can rotate it again. I can rotate it again. And then now we're back to the original square. Right. Do I have a Where's my Sharpie? I have a proper Sharpie nearby. Um, okay. So, nothing too surprising. If I rotate this square four times, we get back to the original square.

[00:05:34]If I rotate four times, um, I can also flip the square. One, two, three, four, like this. I can flip it. I could rotate and then flip and then rotate, uh, etc. It turns out that there's exactly eight ways that you can rotate and flip [snorts] a square. And those are the eight the exact eight um symmetries of the square without like cutting off an edge and rearranging like gluing like regluing or chopping up the square or something.

[00:06:14]um that mathematical object R8, you can treat these like functions or numbers that you can multiply together. Like if I rotate one time and then I rotate three times, 1 2 3.

[00:06:32]Now I'm back to the original square. So there was a rotation and then an inverse operation.

[00:06:42]Um this is the beginning of this is like one of the most standard examples to introduce the concept of a group.

[00:06:52]Yeah. Okay. Let me read chat a little bit and switch over to the iPad. I want this one.

[00:07:01]Um let me read. We got so many views in the last streams. Yeah, the streams have been going crazy. Mostly the video series. El Cuki, you've seen the the series, right? Um the series has been going crazy. It's slowing down now. The series is definitely it's tapering. It's not slowing down to a limit. It looks like it's just tapering and it's it's slowing down, but I think from what I can tell, YouTube is going to be recommending those videos for a while. I don't think YouTube's just going to stop. I think it's going to continue recommending it but to fewer people. Um, is second year of college too late to learn math if I failed math in both semesters of first year? Anyone in the comments can give advice too? Yeah, please always people in the comments you can in the chat you can answer each other's questions and things like that.

[00:08:00]Better late than never. Yeah, I really think it's never too late to study math.

[00:08:05]Um especially if you want to study it as a hobby like you can study math at any point. Um so what is your purpose? Uh cavern engineer enthusiast um if your if your intent is to become a research mathematician I still don't think it's too late. I still don't think it's too late but yeah it's going to be an uphill battle. It might be hard. you have to work your tail off. But you could you could still do it if if you're really determined. But if you if you just want to learn math, like the way that you said it, learn math, you might be like a hobbyist. Like this is the I think that this is the key idea that my stream, my channel has put out into the world and is somewhat unique or at least under represented. obviously so under represented that you can study math for fun. You can study math as a hobby.

[00:09:07]That's what I'm doing. Um, and it's never too late. It's definitely never too late to start studying math as a hobby.

[00:09:17]Um, JC, have you used sum as product or product of sum formulas of trig in calculus yet? Um, no, not in calculus. I mean, not in the my calculus videos. Uh, never too late to study math. You don't need many resources. You literally just need pencil and paper. Yeah, you can watch Math Sorcerer. Or you could watch me. Come on, man. Come on. I'm the new Math Sorcerer, y'all. There's a new There's a new math YouTuber in town.

[00:09:49]Come on. Come on, dude. Um, yeah. You're so thankful that you had good teachers who turned you on to to learning math. Absolutely. You want to see invariance, basically functions that are symmetric by some transformation.

[00:10:04]Oh, yeah. Yeah. Yeah. Yeah. Well, funny enough, we will absolutely be studying some stuff about invariance today.

[00:10:16]Yeah. Hang hang out here with us. Yeah.

[00:10:18]So, we are um dude torch This week we're back to graduate level math. We are going to be studying groups and representations.

[00:10:29]So, you are in the right place.

[00:10:33]Uh, you're doing electrical engineering and math you're bad at cuz your prerequisites aren't good. Well, dude, you well, you you want to sharpen your math for electrical engineering. You need some hella math for electrical engineering. Um, but you can you can also just study it because it's fun. And like, man, um, I believe in you. It sounds like you're not trying to be a math researcher. Yeah, you can keep learning math. Nobody's going to stop you. It is not illegal to study math for fun. Um, it's a little weird, but I'm telling you, it's not you're not going to get in trouble. Hey, what's up, Ard?

[00:11:15]Uh, hopefully your series about basic math is good enough. That's the goal.

[00:11:19]But it's gonna it's going to be a little while. It might be. I don't know. I don't know. Like I want to go slow and I want to work on the fundamentals and the foundations and everything. Okay. So, last time we were here was July 5th. So, let's update this.

[00:11:38]Last time we were working this was July 5th. And today is not July 5th. Today is July 20th, so we are starting to see representation theory a little more frequently. We're now going to do one week high school, one week college, one week graduate school. Um, today I do think that we'll be seeing both sections 3.1 on the basic definitions and first examples and Mashki's theorem and complete reducibility.

[00:12:15]Uh so where were we?

[00:12:20]Ah yeah.

[00:12:26]Okay. So I need to remember what in the world are we doing here? Uh let's just catch up. So this is a study session for me. Um this is not going this is not intended to be a lecture. Um, so I'm going to kind of rip through some of these definitions and remind myself what we have in section 3.1.

[00:12:51]We have basic definitions and our first examples.

[00:13:03]Um we get the definition of a representation.

[00:13:14]We get some examples of a trivial representation.

[00:13:22]An example of a representation of Z mod 2Z. Another example Z mod 4 Z. we get the definition of [snorts] an equivalence an equivalent equivalent representation.

[00:13:43]Um I think I remember that well enough that I could reconstruct it. We get the definition of yeah we get some examples the standard representation of SN. Um then we get the definition of a G invariant subspace. G invariant subspace I remember that uh definition this next one I think I want to rewrite definition 3.1.1 on the direct sum of representations. I want to review that one. I think I kind of have the idea of it, but I do want to rewrite that one.

[00:14:49]Uh, we get some examples. We get a remark about equivalence to the trivial representation.

[00:14:58]Uh we get another example about a representation defined on generators.

[00:15:06]We get a nice table of analogies.

[00:15:09]This is a really really helpful table I think. So I'm going to write it table on analogies between groups.

[00:15:26]uh vector spaces and representations.

[00:15:39]So we compare those.

[00:15:41]Uh then we get the definition of an irreducible representation.

[00:15:50]Oh, Luna.

[00:15:54]Um, I'm sorry y'all. I gotta take a quick break. Um, I'm gonna go get the cat and uh uh go pee real quick. Be right back.

[00:16:22]>> [music] [music] [music] [music] [music] [music] [music] [music] [music] [music] >> Hey. Hey. Look, I got the kitty cat. Hi, baby.

[00:18:27]Hi. Say hello to all all your new fans.

[00:18:31]She's got so many new fans out there now.

[00:18:35]Luna. Luna.

[00:18:39]Okay, she's I don't know why she's been so more shy recently.

[00:18:43]Telling you, usually she's like very very crazy social.

[00:18:49][laughter] Studying maths in rain while having a cat to pet is heaven on earth. What rain? What are you talking about? Um, come on, baby. Yeah, she's a good She's my practice companion. She's my practice companion and she's good. How's it going everybody?

[00:19:09]Uploading your notes. Yeah, I should I should absolutely do that. I just need to find a way to make it easy for me to do to upload my notes. Um, your math teacher is an electrical engineer. Yeah. Yeah. You need a lot of math to to study electrical engineering.

[00:19:29]I think math teacher is a computer science engineer. Cool.

[00:19:35]Galwa theory. Yeah, I dude I want to study some Gowa theory. I've got a book back there that's on the to-do list someday.

[00:19:45]A stray cat. Get her checked. Why?

[00:19:49]What's wrong with her? She's good. She's good. What's wrong? What's wrong with the loon? Okay, come join us.

[00:19:58]Get her checked. What's wrong with her?

[00:20:00]Okay, great. Now we got her joining us.

[00:20:05]Okay. Too many books. I know. I know.

[00:20:13][sighs] Okay. She's too cute. [laughter] That's why we have to get her checked.

[00:20:18]She's too dang cute. It's It's impossible for such for her to be so cute. Okay, let's get back to the iPad.

[00:20:29]Um, so where were we? I'm just catching up and reviewing because it's been two weeks or three weeks depending on how you count since the last time we worked this stuff.

[00:20:44]Uh, although I've been studying every now and then for myself.

[00:20:48]So we have a definition of an irreducible representation.

[00:20:53]Then we get an example.

[00:20:58]Um then we have a proposition 3119.

[00:21:04]Okay, this one is like our first like real result proposition 3119.

[00:21:12]So to summarize it, it says a degree 2 representation is irreducible if and only if um there's no common vector. here to all the group uh representations or the group images.

[00:21:54]Axler is nice. Um I dude, don't get me wrong, I like Axel as a theoretical second course in linear algebra. It's a beautiful book. It's not the book that my class was taught out of. Mine was taught out of um Freedberg Insulin Spence FIS.

[00:22:14]That was a really nice book, too. But so far, I I really like Axel as a reference. It's not your first intro to linear algebra. He says so in the book, too. So, don't listen to people on Reddit. Do not read Axel for your first book. Just don't do it, dude. Like if you look on Reddit, people tell you Strang, go read Strang or go read Axler. No. No.

[00:22:40]Frollay or Lei. Those are your options in my opinion for your first book. For your second book, Axel is a good choice.

[00:22:48]It's not the only one though. There's other good ones for a second book, too.

[00:22:53]Oh, yours was taught out of Insul. Insul is good, man. Insul is really nice. Like I I feel I don't know, man. Like I liked insult. I liked Dinsel. Hey. Hey James, bro. Thank you for supporting the channel, man. I appreciate you very much. Let me give you some credit right here where credit is due. Luna, we got a new Luna member, James Wahberg.

[00:23:33]James Wahlberg 4466. I appreciate you very much.

[00:23:39]Thank you. Um and let's see. First semester linear algebra taught Axel and that explains why you found it so hard. It's Yeah, unless you already have experience writing proofs.

[00:23:57]If you've already learned how to write proofs and you're already interested in abstract pure mathematics, yeah, maybe you could start with Axel. But most people, that's a lot of new info all at once. Like intro to proofs, intro to linear algebra, you know, and an inherent interest in abstract linear algebra. Like, it's just too much, I think. Um, bro, thank you so much, James.

[00:24:30]Yeah. So, if that was your first proof-based class and it was your intro to linear algebra at the same time, I just think that's not the right way.

[00:24:40]Like, were you a math are you a math major and like were you on a were you on like a math major track of some kind? like like was it an honors linear algebra class for math majors?

[00:24:55]Yeah, maybe. But most people learning linear algebra are learning it to become an engineer and that's okay. That is the world we live in.

[00:25:07]That makes sense. Um so I don't know. Yeah, if you if you know that if you're a math major, maybe James Wahberg, 4466, I appreciate you very much. 4466, you're right down here at the bottom of the page.

[00:25:42]supporting uh representation theory.

[00:25:45]Thank you. Okay, so we've got this proposition.

[00:25:50]Um let me see, can I rewrite this so it fits on one line? This is a trick that I've developed for myself to try to like really force yourself to to summarize even sharper like in a Google doc. Can you fit that bullet point in one single line? Like how many words do you need to cut out? Can you abbreviate it enough to fit on just one line? So fe uh I'm going to say degree 2 fee irreducible if and only if fiji share no common vector.

[00:26:44]Ah so close. Okay, pretty close enough.

[00:26:50]I'm just going to say degree 2 irreducible if and only if the group images have no common vector. There we go.

[00:27:02]That's a pretty good succinct little sharp summary. Proofs are scam since the Pythagorean theorem. What are you talking about? What's up high in the sky? How's it going on Twitch? holding it down over there on Twitch.

[00:27:17]One of like two people. Um, okay. It was the non-engineering linear algebra.

[00:27:24]Okay. Which had lots of math majors. I was studying it for econ.

[00:27:28][sighs and gasps] I just I think for Econ you still want the reg I I think you want the engineering linear algebra, but it wasn't the honors version. I feel a first course in linear algebra should only be taught out of Axel if it's an honors course. I feel like he says it in the book. He says it on the back cover.

[00:27:52]So I'm sorry that you had a rough intro cuz linear algebra turned me on to math.

[00:27:57]So I feel strongly about this. Like I love linear algebra very much. Um, and I don't know if I would have if it started out day one as rigorous as uh Axel.

[00:28:09]Maybe. Maybe.

[00:28:14]Um, okay. So, that was our proposition.

[00:28:18]What else do we need? We're catching up.

[00:28:21]So, a corollary of that proposition, I think. Oh, no, no, no, no. We get another Oh, where where did it where did it go?

[00:28:33]Still grew to love it years later. Okay.

[00:28:36]Yeah, totally. I mean, it's beautiful.

[00:28:38]It's a it's a really nice book from what I've seen. Okay. We've got that proposition. We've got an example. Then we have a definition um about complete reducibility.

[00:28:57]completely reducible.

[00:29:04]Um, I can't write this one immediately off hand. I think I know the general idea. It's like it it factors completely into No, I don't remember that one. We got to write that one down.

[00:29:22]Um then we've done this definition on a decomposible representation.

[00:29:37][clears throat] Then we have a lema.

[00:29:51]If fe is similar to a uh decomposible then fee is decomposable.

[00:30:25]I don't like that summary of this. Let's Let's make this sharper.

[00:30:34]Uh decomposable. A representation is decomposable if similar to a decomposible decomp if similar to decomp. Here we go.

[00:31:05]So if a representation is similar to a de to a decomposible representation then it is decomposible.

[00:31:15]Um I understand the statement of that lema. I don't remember how to pro I've proved this one for myself a few times.

[00:31:23]So I'm not going to work that proof.

[00:31:26]Then we have a similar lemos for oh wait when I say similar that's the same as equivalent right similar is the same as yeah I often pronounce that squiggle symbol as um similar but I me I mean to say equivalent Let's make sure that we use the same word everywhere.

[00:32:01]Uh, a representation is decomposible if it's equivalent to a decomposible.

[00:32:09]Um, and then we get the same ideas for completely reducible and irreducible, I think.

[00:32:19]Let me get it.

[00:32:21]Um a representation is irreducible if it is equivalent to a irreducible.

[00:32:41]Another lema. It is completely reducible if it is equivalent to a completely reducible.

[00:33:09]So this is all shorthand like notation or like really just notes to myself right now, right? Uh but it's I think a a good practice technique to be able to write down a succinct summary of the idea in in shorthand.

[00:33:29]Um I mean elements first books look like the theorem was known.

[00:33:34]Yeah. Pie in the sky. You're talking about Uklid's Elements. Um, someday I want to work through that book.

[00:33:46]Yeah, the most famous math book of all time. Okay, now we're going to be moving on. I feel pretty good for the most part about everything in section 3.1.

[00:33:59]There's just like one definition I want to rewrite. Today we're going to be moving on to Mashki's MASH keys theorem and complete reducibility.

[00:34:32]definition on a unitary representation.

[00:34:42]Can I just use fee everywhere to mean representation?

[00:34:48]I think so.

[00:34:51]Um, this one I want to work and write down.

[00:35:00]Um, then we get a proposition.

[00:35:07]Let FE be a unitary representation.

[00:35:17]then it is either irreducible or decomposible.

[00:35:34]Okay.

[00:35:40]All right.

[00:35:42]I mean I starting to lose me because I don't I have an idea of what unitary representations are. It relates to like a unitary uh linear transformation which satisfies the inner product of vw equals the inner product of uv uw.

[00:36:09]But now I'm now it's starting to lose me. Here's the last one. Third in a row.

[00:36:15]Let's pause after this proposition.

[00:36:19]Every rep of a finite group is equivalent to a unitary representation.

[00:36:26]Neat.

[00:36:29]Every fee of a finite group Neat.

[00:37:05]Okay. I have not worked any of this part.

[00:37:10]No, I have worked this one on my own.

[00:37:13]This really cool averaging trick thing.

[00:37:16]I remember reading this last week. It's really cool.

[00:37:22]Um, but I I cannot reproduce this. We're so close to being done with all the main theory.

[00:37:31]I mean, I'm not rewriting all the examples, but corollary.

[00:37:42]I guess I should just get used to writing coralary. CO R correlary says that if FE is a nonzero rep of a finite group Um, it's either irreducible or decomposible.

[00:38:33]Okay.

[00:38:36]Yeah. Okay. So it's important I think that these last couple of propositions and and such are stating that it's a finite group that seems to be popping up a lot more now whereas it wasn't part of the definitions earlier. So got that. And then finally finally the last thing in this chapter is a our first theorem by Mashki.

[00:39:19]Every representation of a finite group is completely reducible.

[00:39:36]Cool.

[00:39:39]Here's all the theorem. Here's all the theory that we're working toward in this chapter. And I feel comfortable about half of it.

[00:39:51]Um but all the starred stuff as we get later like I need to remind myself. So review first the definition of a direct sum of two representations.

[00:40:27]I believe that it's something like fee is Fe1 direct sum fee 2 >> [snorts] >> where each of those are G invariant.

[00:41:04]I think that that's what we want that they're non zero and G invariant.

[00:41:13]I don't think nonzero is part of the definition of direct sum even though it seems like it should be to me.

[00:41:27]Irreducible [snorts] completely reducible a rep G invariant subspace. Okay, suppose that representations F1 and F2 are given then their external direct sum is given by that. Okay.

[00:41:58]Um, I think what I just wrote is something more like complete or like decomposability.

[00:42:12]I think I just wrote down the definition of a decomposible rep. Is that right?

[00:42:21]It's similar. V is V1 direct sum v2 And then you can restrict fee to to each of those complete reducibility.

[00:42:41]Uh-huh.

[00:42:46]I wrote something that's like a little closer to complete reducibility and Decomposable reps.

[00:43:00]What's up, Goku?

[00:43:02]How are you? Good to see you. Um, okay. So, this isn't quite right.

[00:43:12]Um, let's write it out. Suppose that F1 from G to G L V1 and V2 are given.

[00:45:24]Yeah, I feel that we haven't worked with this definition as much as the other ones. We just haven't been using this concept as much. Hey, what's up Divian?

[00:45:37]What's up? How's it going? Uh, you're thinking about internal direct sums, it seems. Yeah, I think I was. I think I was. So, external direct sums. You need two arbitrary vector spaces and then you build an external parent vector space.

[00:45:58]Yeah. So, I need to I need to try to get used to thinking direct sum means external direct sums rather than internal direct sums.

[00:46:09]How's it going, Divium? So, y'all, we are in graduate level math week. We're working one week high school, one week college, one week grad school. And this is a practice session for me. This is me truly living by my ethos that you can study math for fun as a hobby. and I'm learning new math for the first time.

[00:46:31]Um, if you all are interested in watching for the basic math series, that is not this right now, but I hope you enjoy yourselves nonetheless. My goal is that this could be like a study with me kind of radio show almost like a study group. You don't have to understand what I'm learning. I I hope that we can all lock in like uh Zafarb is Zafarb is locking in right now as the kids say.

[00:47:00]Okay, so there is the direct sum deaf.

[00:47:05]So we got that one.

[00:47:09]I keep getting it wrong. So I want to work this example that I keep skipping.

[00:47:22]Let's try to understand direct sums in terms of matrices. Suppose this and this are representations.

[00:47:34]Okay.

[00:47:37]So for example, if F1 um if we have two representations with matrix images.

[00:48:29]Then we can build this external direct sum which maps which represents the group as matrices in M plus M + N square matrices.

[00:49:02]with block matrix.

[00:49:20]So um for any group element the matrix that represents that group element is Fiji 1, Fiji 2.

[00:49:48]Okay.

[00:49:50]Uh more concretely, an example of this is representations of Z mod NZ into uh the nonzero complex numbers and this one would be Z mod M. No, for this one it's both N.

[00:50:40]Okay.

[00:50:44]With F1 of uh G.

[00:50:56]G is a integer mod n E to the 2<unk>i I M / N. And then this other one is E to the -2<unk>I I M over N.

[00:51:30]Then F1 direct sum F2 is this block matrix 2 pi I M / N E to the -2<unk>I I m over n.

[00:52:00]Okay, [snorts] I think I I follow that. And this is a representation that we've seen already previously. So I suppose that's why the book is using that specific matrix. Um yeah yeah yeah it showed this representation um it showed this representation previously in def 317. Okay, keep going. I think that we're done with um reworking that thing now. completely reducible.

[00:52:48]Let's remind ourselves how that goes.

[00:52:54]Hey, what's up Arian Singh Rethori? How are you? Hello. Welcome to my practice stream.

[00:53:03]Um, okay.

[00:53:10]Definition on a completely reducible Um, let G be a group fee a representation is completely reducible.

[00:53:45]If V is the direct sum of um M of N uh subspaces.

[00:54:18]Okay, let's make this a little bit better.

[00:54:40]A representation is completely reducible if the vector space is the direct sum of G invariant subspaces.

[00:55:09]Uh-huh.

[00:55:16]And for each of those subspaces, FE restricted to VI is irreducible.

[00:55:43]Okay. Uh so a representation is completely re reducible if the vector space is the direct sum of G invariant subspaces.

[00:56:08]I mean um f vi being irreducible implies that vi is g invariant right.

[00:56:40]I mean, can't I just say that if it's completely reducible if FE is the internal direct sum of irreducible representations.

[00:56:57]I I I mean that works in my head. Fee is completely reducible if it is the internal direct sum of irreducible reps being invariant equals being a subrepresentation not necessarily irreducible.

[00:57:47]Yeah. Um okay but the right right I noticed that. So then the next point said that fee restricted to vi is irreducible.

[00:57:57]But if ir if it's irreducible then that subspace must be that subspace must be um G invariant right I guess irreducibility a representation um I guess G invariance doesn't make sense for a representation because you couldn't get something outside of the vector space.

[00:58:39]I'm sorry. Be right back. I got to be right back. One more time. BRB.

[00:58:51][music] >> [music] [music] [music] [music] [music] [music] [music] [music] [music] [music] [music] [music] [music] [music] [music] [music] [music] [music] >> Hey. Hey. I'm back. I'm back. What's up?

[01:04:40]So, math canbeahhobby.com. By the way, I have set up that URL if anybody wants to share uh the video or the playlist to those uh to those videos. This is a brand new URL that's easy to pronounce in random conversation.

[01:04:58]Next time that you're talking to somebody on the street and you think that they would like mathematics as a hobby, math can be a hobby.com.

[01:05:18]This is new.

[01:05:23]There's a little bit of housekeeping.

[01:05:27]Uh so if anybody wants to share what we're doing, there's uh there's your chance. Very easy to pronounce.

[01:05:41]All it does is just um redirect to the YouTube playlist.

[01:05:46]Okay. Uh I think that this is an accurate um paraphrasing I think of this definition a completely a a representation is completely reducible if it's the direct sum of irreducible subrepresentations.

[01:06:16]Maybe that's a be better way to put it.

[01:06:19]of irreducible subreps and then we already get internal clarified by the idea of subrepresentations. I think that this is accurate. Can somebody check me on that? I I think that this is correct.

[01:06:48]I don't remember. I think that the book has defined a sub representation, but it wasn't in the long list some before, but I swear it's used the word subrepresentation somewhere.

[01:07:08]Yep. Okay. Let fee be a representation.

[01:07:12]If w is a g invariant subspace, we may restrict fee to obtain a representation fee restricted to w by setting that that uh precisely because w is g invariant.

[01:07:24]We have fgw in w sometimes one says a fee restricted to w is a sub representation of fee.

[01:07:37]Yep.

[01:07:51]Yeah, there's this little bit of the book that I keep kind of my eyes keep glazing over and like kind of glossing over. Um, but now I think I'm understanding a couple of paragraphs better. Hey, what's up? Say stuff. How are you? Yeah, it seems right. Yeah, man. Um, hey true is Isatia. How's it going?

[01:08:18]How's it going? I didn't pin this message correctly. Um, so y'all I think that you want what you want is the Discord. You can join the Discord and you can join the study group.

[01:08:33]There's a study group channel for basic mathematics.

[01:08:38]Um, you're all right. A little tired.

[01:08:40]Yeah, I feel you. I'm like ready to go, man. I've been excited for this week.

[01:08:45]We're back to graduate level math this week for my practice sessions. Don't worry, we're still continuing the Math Can Be a Hobby series. Uh, in fact, I plan on filming episode 4 later today.

[01:09:00]So, every week I want to I want to make sure that um I can point people to this. First off, we have a brand new URL.

[01:09:11]Um, we have a new structure that's forming which is two videos per week.

[01:09:31]We've got one theory video and one examples video.

[01:09:44]Uh today I will be filming theory for section uh 1.3 on rules of multiplication.

[01:09:56]Uh I should also say a little bit better structure that is forming. I think this will work well.

[01:10:08]One section per week.

[01:10:17]One section per week. Two videos.

[01:10:21]um one is on theory, one is on examples and then the weekend Saturday discussion group discussion actually there's two different discussion groups one is a live stream and then one is a zoom call um so this structure is forming and Today I will be filming my theory video on uh multiplication.

[01:11:03]I need to learn latte to type out some of your thoughts. I need to add latte bot to the discord.

[01:11:12]Oh, and somebody else asked, uh, how can they how can they talk about math stuff? Um, join the Discord link in the description, link in my bio.

[01:11:28]Uh, but discord.jpractices.com, jcpractices.com.

[01:11:37]That's my hacky way to have a free invite URL cuz usually you have to pay to have like discord.gg/jcpractices.

[01:11:48]You got to pay like $20 a month or something for that. Nah, screw that.

[01:11:53]I've got my own URL.

[01:11:55]discord.jcpractices.com just redirects you there. Ha, free.

[01:12:00]Well, almost kind of. I still have to pay for the URL, but I use it for like 100 redirects like this, so no problem.

[01:12:11]Um, yeah, prerequisites for representation theory. Technically, it's just linear algebra and and groups, but it's I would say mathematical maturity, proof writing skills. You have to be good at proof writing. Um, like this book is is not easy. This book is dense. It takes sometimes a week to get through a page.

[01:12:39]It's like that kind of math book. It's it's different. So, don't worry if you're not, you know, following everything here. This is me learning for myself. I don't want to like over like make it sound too scary. Like, you can learn this. Like, yeah, you can learn it. It's not, it really isn't too much prerequisites.

[01:13:03]Um, yeah. If you're having trouble when you when you say just looking at the linear algebra review, do you mean from this book? Linear algebra review from this book. It's it's dense. It's actually very dense. Um, but you know, the the book doesn't get easier if I have a I I don't know how howie how is another book that's recommended for undergrads.

[01:13:36]Um that one might be friendlier. I don't know. I really don't know.

[01:13:41]Oh, from the book looking at the notation. Yeah, this book is it's a it's taking me some time and I feel you know I feel not bad. I I feel decently good about my math background.

[01:13:56]I don't feel like this is an easy book.

[01:13:58]I feel that how might be easier. Have you worked have you have you learned abstract algebra yet? Have you learned group theory? Because I would highly highly recommend that you work group theory before representation theory. Um, okay. So, what's the other thing, y'all?

[01:14:18]I now have affiliate links. I finally did this.

[01:14:24]affiliate links.

[01:14:27]Um, so if anybody wants to get these books, let me know. I already have one set up in the video descriptions for basic math.

[01:14:42]Um, and more soon.

[01:14:50]So, let me know if anybody plans on getting one of these books and I will pause and and set up that affiliate link. Um, I'd appreciate the dollar that I get, of course, but partly I also just want to make sure I get three in six months or whatever so that they can so that they'll let me keep my account. You have to hit like a certain number per month or per every six months in order to to not get deactivated.

[01:15:20]Oh, you have done a course on abstract and linear algebra. Okay. Well, then I think that you're Yes. Uh Arod is has a good point. The review chapter in this in this book is just like rapid fire. just like here's everything from uh from linear algebra.

[01:15:42]Let's just like summarize all the most important stuff without proof. Um so I wouldn't worry too much about it. If you want to practice your linear algebra, it sounds like you're ready for Axel. Like get a just get a copy of Axel and just work some problems, dude. You can do them. You can do them just day by day.

[01:16:06]Okay, so there's a little bit of housekeeping. Um, so now I've got a definition of complete reducibility and it's in some words that are going to stick in my head a little bit better. A completely reduc representation is uh the direct sum of irreducible subrepresentations.

[01:16:34]Yeah. Okay, we got that one.

[01:16:41]Next, I wanted to write down the definition of a decomposible representation. These concepts seem so similar, and I guess it's cuz they're equivalent.

[01:16:53]Um but a decomposible representation.

[01:17:08]Ah not necessarily irreducible. Now I think I'm starting to get the the distinction a little bit better.

[01:17:19]So the definition of a decomposable representation fee uh is de is said to be decomposible if [snorts] If the vector space is the direct sum of G invariant subspaces, but these subspaces are not necessarily irreducible like the the restriction is not necessarily necessarily the subrepresentations are not necessarily irreducible.

[01:18:26]um G invariant nonzero subspaces.

[01:18:43]It's annoying to always say nonzero g invariant nonzero subspaces.

[01:18:53]Okay.

[01:19:01]Okay. So the subrepresentations are not necessarily irreducible.

[01:19:12]Okay.

[01:19:14]Then um rewriting this lema one more time fee equivalent to a decomposible then it is decomposed. osable.

[01:19:58]If a representation is equivalent to a decomposible representation, then it is decomposible.

[01:20:07]Okay.

[01:20:13]Uh we got another lema.

[01:20:17]If is equivalent to a irreducible then it is irreducible. Oh, if fe is equivalent to a completely reducible Then it is completely reducible.

[01:21:15]Yep.

[01:21:18]I'm not going to work the proof because I've done that on my own a few times.

[01:21:22]But we Let's cross these things off of our checklist.

[01:21:27]We got the definition of a decomposible rep.

[01:21:31]Now we're starting section 3.2.

[01:21:34]Hell yeah.

[01:21:36]Yo, bro, you're thinking of streaming math. You should. You should. We need way more math streamers in this world, I think. Just give me give me a little bit of credit and like, hell yeah, dude.

[01:21:49]That's awesome. Just give me a shout out every now and then. Uh, if if I if I'm your inspiration to start streaming, that would make me so happy. That would make me so happy, dude. Um, after graduating in your free time, I mean, how long away is that?

[01:22:11]Uh, hey. Hey, Lock. What's up, bro? Yeah, I mean, you are and you're not. Okay, so we got all the way through ch section chapter 3.1 basic defs and examples. I think today's the first stream where we're going to spend significant amount of time on Mashki's theorem and complete reducibility.

[01:22:37]Uh so I I'm really just catching up reviewing what we've done in the past in 32 we've got Mashki's theorem.

[01:22:49]Ah next semester. Okay, let's go. Dude, why not just start now?

[01:22:54]So, what do you all think? I'm so curious.

[01:22:59]Why are the bullet points so compacted here? It's because like this was meant to be like the most highlevel just reminding myself of what all everything is and starring the ones that I wanted to actually rewrite and review.

[01:23:20]This is a a practice technique in my opinion that's always been very helpful to me like at large companies when you want like a summary of a meeting or something or you want like a an informal document that captures a lot of ideas like it has to fit in one bullet point line. like fit that idea in one line, you know, and and use as many abbreviations and, you know, paraphrasing and incomplete grammar to make it fit on one line. I don't know that I've always thought that that helps. Uh, yeah, completely reducible. Yeah, ex. You got it. You got it. Completely reducible.

[01:24:11]I mean, it made sense when I was saying it out loud and I was writing it and and the point is me practicing. It's not for somebody else to read this. Uh, hey philosophies, can you lunch like playing?

[01:24:28]What do you mean a playlist? Do you want a playlist of my music or a playlist of my videos?

[01:24:37]Actuary exams. Um, yeah. So, here's here's what I was thinking. I was just going to say say stuff also. Let me let me highlight. We got we got a brand new URL. Everybody, math can be a hobby.com.

[01:24:55]So, if you want to share my my videos, my playlist uh with anybody, there's your opportunity. Next time you're talking to somebody at the at the grocery store or at the cafe and you think that they would like mathematics, tell them mathbehobby.com.

[01:25:17]Um, okay. So, let me answer your question or your your comment. Um, could somebody do this while they're a student? Could somebody stream themselves working homework exercises or would that be like academic dishonesty or you know like would that be cheating? Would you get in trouble as a student if you were streaming yourself working your homework exercises and getting help from people on the internet? Like, you know, would a university shut that down really quick as like copyright infringement? Like, I'm writing a lot of stuff almost almost verbatim from the books. Like, would would streaming your homework solutions be frowned upon as a student? Ah, maybe.

[01:26:20]I think maybe. That's where I've got an advantage because I'm a hobbyist. I've got an unfair advantage. I can't get kicked out of university for doing this, right? I think it's I think it's an interesting idea like question like what happens if I inspire somebody to do what I do and then the and then their university gets mad.

[01:26:43]What do what happens then?

[01:26:51]So that's like what say stuff is saying like say stuff wants to stream actuarial exam prep but if you show that on on screen it's it's copyrighted and they might not be happy about that.

[01:27:05]Um good luck with your GA write up Lockheed sounds interesting. You want a whole math course man? I mean I'm just 18 and I really want to master math which is impossible. just a course playlist with everything I need literally. I mean, it's brand new. It's brand new and we haven't done anything more than addition, but what you want is my new playlist. Math can be ahobby.com.

[01:27:31]Literally, this URL, this is a YouTube playlist.

[01:27:37]It's just a it's just a fancy link.

[01:27:44]this is the playlist that you're looking for for basic mathematics.

[01:27:53]I mean, even listening down all the topic in math is a hard task, let alone a playlist. Yeah, it's going to take a while. It's going to take a while. But if you want to just relearn math or get better at math, this is starting from the ground up. This I think that this is what you're looking for, dude. Like, seriously.

[01:28:16]like we are starting from the ground up and we're going to learn all of high school math, but we're going to learn it the right way slowly and in depth with proofs.

[01:28:27]Today I'm going to introduce multiplication. Like last time we talked about addition like that's how like we're starting from the basics but we're doing it right. We're doing it better, deeper proofs, understanding everything conceptually, like really conceptually.

[01:28:49]Math subject classification 200 pages.

[01:28:52]No, I haven't seen that. You could do the free sample. Yeah, see that's what I'm thinking. Like I've done I felt a little nervous about it, but I've done streams working and showing the entire uh sample exams from the college board for the for the SAT. Like I worked the sample SAT exam and I worked the sample um AP pre-calculus exam.

[01:29:23]So, uh, yeah, like I don't know. I I think it I have a hard time seeing these publishers or exam companies getting mad about streaming their free sample materials.

[01:29:46]And dude, I think a lot of people have bought this book already. Like a lot of people have already bought this book on basic math because of what I'm doing. So I I have a hard time imagining that I'm gonna get in trouble. But if I do, then we'll figure something else out. I'll I'll just have to rewrite every problem with my own numbers instead, which is a pain in the butt.

[01:30:11]I think everything I'm doing is very fair use. I think it's I think everything I'm doing is super super fair use. Yeah.

[01:30:25]Yeah. Okay. So, let's keep going. Um, we've written out reviewed all the definitions and lemmas and I've worked Dude, I kind of want to tempted to I've done this one on my own so many times.

[01:30:45]I've done these on my own a bunch of times, so I'm not going to I'm not going to rework the proofs of this one and this one, but I've done those on my own. So, let's keep moving.

[01:31:01]Let's get to Mashki's theorem. Hell yeah.

[01:31:26]Okay. So we need the definition of a unitary representation. now uh which I imagine relates to the concept of a unitary transformation in an inner product space. So we're going to have to say that V is an inner product space and that the I think it's going to be V is an inner product space and V the inner product of VW is the same as the inner product of F VW for any group element G I think that this is it.

[01:32:26]So let's find out.

[01:32:29]Yep. Okay. So precisely let V be an inner product space.

[01:32:41]Um and FE from a group into the invertible transformations of an inner product space is a unitary representation is said to be unitary. Very if Fiji is unitary for all group elements.

[01:33:35]I'm not going to rewrite the definition of a unitary transformation, but I think [snorts] a college entrance exam, math and science, two subtests of the for basically fundamentals and more wide range of topics than deep how to prepare I don't know um okay good question uh at remminis great question I love taking this question it's a very good one reminis Um look into the concept of breadth first search and spaced repetition. Look into these concepts.

[01:34:52]Repetition.

[01:34:56]Um in breth first search in algorithms you visit every node before you move on to the next layer in a tree.

[01:35:24]Um, [snorts] in math, the way I see that is you work it one, you work one exercise from section one, then you work one exercise in section two, then you work one exercise in section three, and you keep doing that until you get stuck.

[01:35:50]And if you get stuck, I think that you should reset. If you if your purpose is to relearn or or see a lot of math and like get a breadth and prepare for an exam with a lot of stuff on it. Do that until you get stuck. Then reset and then work the second problem from every section.

[01:36:17]And then this time around, maybe you're still stuck on that problem. Maybe not.

[01:36:25]You know, maybe you reset again and you work the third problem from everything until you get to that problem. And then at some point, you're going to freaking be able to work that problem. Don't forget to reread the sections and work the examples from the section. uh but keep resetting and work previous problems from previous sections and then just keep going.

[01:36:54]uh you know once you've if you can solve that exercise move on to the next one work that problem keep going and just keep adding more and more material until you get stuck and then reset again and work everything all the way up until then. Um, this method is going to get you like at some point.

[01:37:25]It's not like you should spend a whole hour or day just on one problem for everyone.

[01:37:31]Some point these earlier topics are going to get super easy and you can just rip through one. You know, you can get through multiple all in one day.

[01:37:46]Um, yeah. I hope that that helps a little bit. So, the way I'm doing this is part of the way I'm doing this is I'm doing a week of [snorts] high school math.

[01:38:10]I'm doing one week of college math, bachelors of science, and then I'm doing a week of graduate school math, really like a master's degree, and then I reset.

[01:38:36][cough] And the purpose is different each time.

[01:38:41]In high school math, I'm the purpose is more teaching and being able to explain the concepts and getting through a lot of concepts, but more about the presentation of teaching. In college math, it's a bit of teaching. It's a bit also of re, you know, for me remembering and relearning some math like abstract algebra.

[01:39:07]or something. But right now, we're in the graduate school math and I am actually learning this math for the first time.

[01:39:16]So, I'm not the purpose is not for today teaching. Um, hey, what's up DZ John? Hi, how's it going? I'm working representation theory of finite groups.

[01:39:32]This is uh a subject about studying symmetry using linear algebra. So group theory is this really beautiful field in abstract algebra.

[01:39:44]Uh it kind of takes the idea of vector spaces and supercharges that to like even more abstract I more idea u more abstract settings. Um, and it turns out that linear algebra is super useful for understanding um, abstract algebra even more. Yeah. You you like it. Awesome, dude.

[01:40:11][snorts] Yeah. Heck yeah. Yeah. You worked dumb and foot. Well, then you could definitely, if you've worked dumb and foot, then you're ready to join me for this topic. Um, did you work the representation theory chapters in DNF?

[01:40:34]We should also add in some DNF exercises in this stream.

[01:40:40]One day of representation theory, but geometric algebra. Uh, yeah. So, it sounds like you have learned some rep theory. I am not at characters yet. No, I am working um I am working out of Steinberg and I'm I'm slowly sipping it like a fine like a fine wine or a fine scotch. Uh I'm in section two right now on um Rep theory uh by Steinberg is the book that I'm using.

[01:41:21]Um, in high school math, we're doing this series that is blowing up. Basic math by Lang.

[01:41:35]College math. I don't know.

[01:41:38]Number theory.

[01:41:41]Number theory or intro to proofs. Um, no. Intro to proofs. number theory, abstract algebra, maybe dominant foot, or maybe gallion.

[01:41:58]Yeah, Lockheed wants wants uh number theory. Yeah, characters are next next chapter. That's right. Oh, you were a teaching TA for uh a bachelor course, dude. Well, you found the right place.

[01:42:11]You can help me. I I mean, it's been what year is it? 14 years since I graduated. So, I am dusting off cobwebs and I'm learning I am learning this for the first time myself. So, heck yeah.

[01:42:28]Yeah. Arod Arod, DZ John, what do you all think? Like, what should be my college math topic? because we have a lot of awesome newcomers who are following the Lang basic math series.

[01:42:50]So, you know, I I want them to als like like some people might want to see more advanced proof like intro to proof math.

[01:43:01]It could be Axel, dude. It could be Axel, proof oriented Axel. I know that some people are annoyed at how much linear algebra I've done. Um, number theory is very tempting. It could be number theory. It could be um, could be number theory or it could be intro to abstract algebra. Uh, I don't know. I'm very curious what you all think would be best.

[01:43:31]I'm leaning number theory or abstract algebra. So, I mean, what do you think, John? because you've worked dumb and foot.

[01:43:40]Um my my university had intro to abstract algebra and proofs.

[01:43:54]Like our intro to proofs course was with abstract algebra and it was taught out of baby hungerford like undergrad hungerford.

[01:44:08]I liked it, but it's a it's kind of a weird rings first approach, and I don't have my copy anymore. Um, and then there was a year-long sequence for, you know, people who wanted to go to grad school. two semesters of uh this was like the the big boy abstract algebra and it was taught out of dum it and foot.

[01:44:43]Um I don't know if I would have done as well in dominant in in working dominant foot without an intro to proofs an intro to abstract algebra. So, I've heard good things about Gallion here, and I actually just got a copy of Gallion. Um, number theory is very tempting. So, I think I feel like a friendly intro to abstract algebra could go here or intro to number theory, often called elementary number theory.

[01:45:32]Uh, I just don't know the best the the right book to use for this. I have a copy of Andrews, which is a really nice little Dover book. Um, I also have an old copy of Rossin, like the undergraduate rosin, not the graduate level Ireland rosin. And in fact, they're not even the same rosin, which is really fascinating.

[01:45:59]But yeah, what do you all What do you all think would be best? And me, too. I've wanted to work a Luffy. I really want to work a Luffy.

[01:46:14]Uh, dominant foot was your first algebra book. Hell yeah, dude.

[01:46:21]Me, too. I'm thinking cominatorics would be really another option like discrete math and combinotaurics cuz I've never learned that myself. But then I think I'm mixing purposes cuz this series right now I'm learning one new subject for myself.

[01:46:40]Um, I just I think that the book might rotate in the college math series.

[01:46:48]Um, Cominotaurics is very tempting to me, too. But I think any of these would be better than like Velamin, like something that's just a pure intro to proofs book.

[01:47:00]I I don't think that's the right way to learn proofs. I think you should learn proofs in context of math, not just proofs for themselves. I'm very tempted to to read a Luffy, but if I did a Luffy, I think it would a Luffy's book, which I have back there and I want to work through a Luffy's book is a graduate level book.

[01:47:26]He has an undergrad algebra book too, but people don't talk about that one as much. His category theory book is an alternative to dumb it in foot and it is a category theoretic approach treatment.

[01:47:44]Um, and yeah, I don't know the best like the best number theory book for undergrads of elementary number theory. Ireland and Rosen is a great graduate um level book.

[01:48:05]It's a great GTM book. Uh, and it's called a modern introduction to elementary number theory or something like that like a or to classical number theory or something. But it's like a pre-alggebraic number theory.

[01:48:24]Uh and it's elementary number theory, you know, assuming mathematical maturity like Ireland and rosin is really really nice.

[01:48:38]And I I worked that a while ago for a bit uh for funsies because I'm a nerd.

[01:48:45][laughter] I'm a nerd. Um, abstract algebra is a nice setting for more category theory. Yep. And channel mathematical adventures has self-studied all of Gallion and done all the exercises. It's all on YouTube. Yes, that's what I'm thinking. I I'm thinking I could like kind of follow what he's done. I don't know if he's published all of his exercises, but he's shown that he has worked the entire book. And I think that's freaking awesome. like I, you know, it's not like he like he's done it and nobody else is allowed to ever work Gallion again, but I think I've I've been turned on to the idea of working Gallian because of his channel. Like he really likes it and I know now he's working dominant foot. Um, you like Frolle for abstract algebra. I love Frrale for linear algebra. So, I bet I would like it for Yeah. Did everything before abstracts get renamed to elementary? Kind of actually. Yeah. I think that's accurate.

[01:49:48]I think that's accurate. Lock. Yeah.

[01:49:51]Visual group theory. Oo. Yeah. There's just like thousand group theory books.

[01:49:58]Cominotaurics is too many numbers.

[01:50:01]You're scared of numbers.

[01:50:03]For this reason, I'd much rather number theory. Okay, I would love intro to proofs because high school geometry teachers struggle with getting students to understand how to write a proof. It might dovetail with a bit of a dive into Uklid. Uklid would be fun. Um, my intro to proofs I want to be doing with Lang with basic math intro to proofs.

[01:50:31]I know that we're not going to see all of the same proofs that some of the other books do and some of the proofs need to be handwavy for this book, but like this this is the series that I have as an intro to proofs. Um, and it's it's freaking it's nice. Like this book is really nice. Basic Math by Lang.

[01:51:02]And yes, High School Geometry is just the worst intro to proofs. I think I think this book is such a better intro to proofs and just like some like little bit of number theory. like a little bit of number theory and this stuff like I wish I could just rebuild the high school math curriculum in the US. Like I don't think we should be teaching, you know, axiomatic geometry. I just don't think we should be teaching axiomatic geometry at all. I'll say it.

[01:51:38]I think it should be offered as like an elective, but it shouldn't be part of the main sequence.

[01:51:45]Aimatic geometry, I'm going to say it like it's just not that relevant.

[01:51:55]uh proofs in the axiomatic method are incredibly important and relevant and everybody should learn that but we should be learning it in the regular math that we all learn like not in aimatic geometry. Alvaro Lozano Robo made a book on Yeah, that guy's awesome.

[01:52:15]I'm a big fan of what he's doing. He's got some really cool videos. I emailed him and he didn't respond to me. I'm sad. Um, yeah. Yeah. Other way around. Classical introduction to modern number theory.

[01:52:31]That's what I meant to say. Classical intro to modern number theory. So, it's like it is an elementary approach, but guiding you toward algebraic and modern number theory. But it's an elementary number theory book.

[01:52:50]elementary methods guiding you to modern number theory.

[01:52:56]Yeah, I already know algebra. So like the purpose for me working Gallion would be teaching and introducing others to algebra.

[01:53:09]Um I have worked dumb and foot and I've worked some of Lang. Like I've I've worked lang a bit. Like if it were me, I would love to add in like commutative algebra and add in um what is it?

[01:53:23]Eisenbud back there. Eisen I have Eisenbud with a view toward Yeah. Commutative algebra with a view toward algebraic geometry. Like I would love to work that but yeah dude that's so validating. There's so many people who say that that like you hated math because of axiomatic geometry. You also taught you also learned aimatic geometry in in India.

[01:53:55]And I it's because that was the most successful math book for thousands of years. Um but it's just it's been watered down so much that we've lost the plot. We've lost the point of it.

[01:54:16]Anyways, let's keep going.

[01:54:20]Like, I haven't looked yet. I have not looked yet, but I would be shocked. I would be completely shocked if Lang does not have Uklid's proof of the infinitude of primes in this high school math book.

[01:54:40]Yeah. like I would be shocked and I I'm confident that he must have the inductive proof for g for um the sum of integers like he he has I'm confident that induction is in this book that um freaking you know the the proof of the irrationality of the square root of two like proving that square<unk> of two is irrational. Like these are all proofs that I learned in my intro to proofs courses in college. We should be teaching that to high schoolers. We should be teaching that to high schoolers.

[01:55:27]Oo, Weissman number theory. Haven't heard of that one.

[01:55:31]Eisenbud is a nice reference.

[01:55:35]I want I want something friendly. Miles Reed, that sounds like a good book to me. Undergrad Commutative Algebra. That sounds sick.

[01:55:46]Um, I've had a total of 10 seconds of fun in high school geometry when a couple friends and I realized a setup from a question had some more nice properties being true than the question asked for.

[01:55:59]Yeah, it you only enjoyed 10 seconds of your high school geometry class. That is really funny.

[01:56:11]Yeah. Anyways, um, cool. Unitary representations. Let's get back on topic. Uh, thanks everybody.

[01:56:21]I'm, you know what? Okay. I'm sorry. You all, please chat amongst yourselves.

[01:56:27]Introduce yourselves. Tell us where you're where where are you all writing from. Who? Oh, and this is Divium, too.

[01:56:34]Yeah, that's right. It's Divi. Good to see you, dude. Um, also I call everybody, dude. Please let me know your pronouns. Um, you know, don't mean to assume any genders or anything like that. Where are you all writing from?

[01:56:50]What kind of math are you studying? Um, what are you looking to get out of this channel? Um, and just chat amongst yourselves. I'm going to put away chat for like 20 minutes or so, 10 minutes maybe. If Hawaii is up, I am studying.

[01:57:06]I'm not reading chat, but you all can can chat amongst yourselves. Um, I want to get some work done. I want to lock in. I want to get to Mashki's theorem today for the first time ever. Um, all right. So, we got the definition of a unitary rep.

[01:57:22]Now, we've got a new proposition.

[01:57:26]Um, who cares what the numbers are, but whatever. Let's write it anyways. Prop 323.

[01:57:40]All right.

[01:57:43]A unitary representation is either irreducible or decomposible.

[01:57:50]Let fe assume finite group in this proposition.

[01:58:11]It looks like then fe is either irreducible or decomposible.

[01:58:36]Hope you all don't mind me putting away chat every now and then. Um, I worked this proof once on my own and it was really pretty and really cool.

[01:58:47]So, I want to work this one again.

[01:58:55]Oh, wait. No, it wasn't this one. It was the next one that I thought was really fascinating.

[01:59:02]Let's work these proofs.

[01:59:06]After I just write down all the statements, I want to make sure we actually get to Mashku's theorem today.

[01:59:13]Prop 3 2 4.

[01:59:20]Every representation of a finite group is equivalent to a unitary rep.

[01:59:29]Every rep of a finite group is equivalent to a unitary rep.

[01:59:52]I like this notation to say finite group.

[01:59:58]I don't know if that's annoying to people, but there you go. Every representation of a finite group is equivalent to a unitary rep.

[02:00:09]Cool.

[02:00:19]Wow. All right.

[02:00:23][snorts] That's really cool because this says the representation itself does not need to be.

[02:00:37]Do we assume that we're in an inner product space everywhere? All right.

[02:00:41]Have we already assumed?

[02:01:02]Huh?

[02:01:10]Yeah. No, we don't assume that V is a inner product space in this one.

[02:01:30]That is so damn cool. Yeah, that is ah that's super cool to me. All right, corollary.

[02:01:40]I will work this proof. This one I think is going to be important for me. I want to really memorize and understand this proof to do.

[02:01:53]I've worked it privately but not on stream yet. Corollary uh 325 FE is a nonzero rep of a finite group.

[02:02:33]Then fee is either irreducible or decomposible.

[02:03:11]Every representation of a finite group is equivalent to a unitary rep.

[02:03:29]H because every unitary rep is either irreducible or decomposible.

[02:03:46]So if every rep of a finite group is equivalent to a unitary rep and every unitary rep is irreducible or decomposible, then every uh rep of a finite group is either irreducible or decomposible. Yeah. So this one just connects the previous two propositions together.

[02:04:09]Yeah. Okay. I think I see that.

[02:04:19]And then finally we get to the statement of the only theorem in this entire chapter.

[02:04:29]Theorem 328 by Mashki.

[02:04:39]This is our big main result.

[02:04:49]Every rep of a finite group is completely reducible.

[02:05:16]Okay. So every representation of a finite group is completely reducible. Do we get that every rep of a finite group is equivalent to a unitary rep and every unitary rep is irreducible or decomposable.

[02:05:44]If it's irreducible then it is completely reducible automatically. If it's decomposable, I think we use these lemas.

[02:06:19]either irreducible or decomposible.

[02:06:28]So I suppose we show that [snorts] I suppose that we show that every representation every irreducible representation is completely reducible and every decomposible representation is completely reducible.

[02:07:03]Is that true? Every decomposible is completely reducible.

[02:07:18]Okay. Wait. So, decomposible has the vector space um split into G invariant subspaces, but those subspaces are not necessarily irreducible.

[02:07:46]But every Completely reducible requires a direct sum of irreducibles.

[02:08:54]Oh. Um so every if every uh yeah this is like it's it's really similar to uh the fundamental theorem of arithmetic how every integer can be factored into primes or you can you can find a factorization of any number right 2 * 26 or um so 2 is prime 26 is not prime but it is decomposable into primes right and then so you it keeps the vector space keeps splitting. Um where some of them are irreducible and others are not irreducible but they are decomposable.

[02:10:18]So then each of the so it's kind of by induction on um [snorts] the dimension of the vector space I think because every time that a decomposible vector space splits into two Then assuming that they're nonzero subspaces then we are reducing the dimension every time in the subspaces.

[02:11:05]So eventually this keeps splitting down by using the fact that every every sub representation is either irreducible which is equivalent to primes prime numbers or decomposible which is equivalent which is similar by analogy to like a a composite number.

[02:11:46]Then every time that we do this factorization, you reduce the um the size of the vector space and ultimately you need to factor into irreducibles.

[02:12:06]Okay, I I think I see the path on how we get to Mashki's theorem.

[02:12:14]Okay, cool. Hell yeah. Let's And then that's it. That's the That's That's it.

[02:12:24]Um I've worked a definition on unitary reps previously [snorts] on my own. I think I remember this stuff.

[02:12:39]A crucial fact which makes unitary rep so useful is that every indomposible unitary rep is irreducible as the following prop shows. Okay. So, let's work the proof of this of this bad boy prop whatever whatever fee unitary implies irreducible or uh decomp decomposible.

[02:13:24]We do not need finitness of the group for this one.

[02:13:38]I know prop and proof look quite similar, especially with my weird handwriting.

[02:13:45]Um all right so let me try on my own.

[02:13:50]Suppose fee is a rep of a group into a vector space and let's assume that it is unitary.

[02:14:18]Then um if I want to prove that it's either irreducible or decomposible Then we can [snorts] take a let's take a subspace of V.

[02:15:34]fee restricted to W is either irreducible or not.

[02:15:49]Suppose v restricted to w is not irreducible.

[02:15:59]So there exists um a nonzero subspace W prime.

[02:16:26]Oh wait, no we want fee itself is either irreducible or not.

[02:16:45]Um suppose it is not irreducible. So there exists a G invariant.

[02:17:03]If there exists a G invariant [snorts] subspace strictly between zero and V.

[02:17:28]Then we want to show that um V is the direct sum of two subspaces for some V1 and V2 subspaces. Uh, decomposible does not require invariance, right?

[02:18:16]Let me remind. Oh, no, no, no. They do need to be G invariant, but they do not need to be Okay. Yeah. Yeah. I think I'm getting the analogy with prime factorization a bit better. Um, okay.

[02:18:29]So, we want we want to show that we can find G invariant subspaces V1 and V2 and they need to be non zero.

[02:18:53]>> [snorts] >> Okay.

[02:18:58]So, there exists I mean I already see where what the one that we're going to choose. Uh I'm pretty sure that our first subspace is going to be W and the second subspace is going to be WP.

[02:19:16]uh which exists because V is an inner product space and V inherited that property by the fact that we know that FE is a unitary rep. V being unitary implies that the vector space is an inner product space and an inner product space always uh has a direct sum demp of any subspace and its orthogonal complement.

[02:20:14]Um assuming that FE is not irreducible W is non-trivial and then therefore W per is non-trivial.

[02:20:32]Hell.

[02:21:05]And it implies that W is G invariant. We show um W per is G invariant.

[02:21:24]Let um W prime be in W per G be a group element.

[02:21:42]Then we want to show that FGW perp is in W complement or yeah FG W prime is in WP.

[02:22:29]I just don't see where we've used unitary yet or how I will Thus if fe if we assume fee is not irreducible then it must be decomposible and we are done. I just I don't see where I've used uh unitary anywhere.

[02:25:14]I've only used inner the fact that it's an inner product space.

[02:25:22]So something feels wrong.

[02:25:47]Um, okay. Wait. Okay.

[02:25:51]Do I need to show I think I think I'm cheating here.

[02:26:23]No. Well, wait.

[02:26:26]Am I cheating here?

[02:26:31]because I don't know what I I don't am I I was going to say I don't know what field we're working over. We don't have any assumptions over the vector space itself.

[02:26:58]We just have Fiji is a linear transformation an invertible linear transformation of V into itself.

[02:27:14]But if it's linear then we have this property right and okay I don't know I see I think I'm I'm cheating right here.

[02:27:51]Assuming that V2 is not in capital W.

[02:28:00]Yeah, I'm cheating right there. That doesn't this this step doesn't work.

[02:28:08]Okay, I got I got the basic setup of this thing correct. Let's try it again.

[02:28:14]And this time I'm going to read the proof. Um, essentially I need to show that the inner product of two things are zero so that therefore one of them is in W per.

[02:28:35]Let's do it again.

[02:28:41]F unitary implies irreducible or decomposible Suppose that FE is not irreducible with a non-trivial G invariant subspace W.

[02:30:26]Then V is an inner product space. So it has a direct sum decomposition.

[02:30:34]And we show that WP is um G invariant.

[02:31:02]>> [snorts] >> In other words, to show that it's G invariant, the image of any vector in WP has to remain in WP.

[02:31:45]Okay. So let WP be in or W prime be in W per.

[02:31:54]So and I should say nonzero.

[02:32:08]So v = w + w prime or w prime is v minus w for some Then if I can show that the inner product of VGW prime with a vector And Okay. So Fijiw is some W2 with W2 in capital W as capital W is G invariant.

[02:36:09]Right?

[02:36:12]So, Therefore, Fiji W prime must be in W because the inner product of it and some other element in capital W is zero.

[02:37:12]I wonder if I need to.

[02:37:21]I don't know.

[02:37:24]So therefore, WP is G invariant.

[02:37:37]And so FE is decomposible.

[02:37:45]Q E D.

[02:37:48]I think if V is in WP and W is in W then Okay, I got something extremely similar.

[02:38:25]I got something very similar. I want to keep moving on. I'm Mine's a little bit different, but it's I think it's the same. I think it's ultimately about the same. Let me catch up with chat. What's up, everybody?

[02:38:56]Hey, happy Abe. What's up, bro? Good to see you. Thank you for tuning in, Eisenbud. Okay, I remember these comments. How's it going, everybody?

[02:39:11]I'm reading chat again.

[02:39:14]Uh, can you say where you are in Steinberg?

[02:39:20]I'm in 3.2.

[02:39:22]Um, Nashki's theorem.

[02:39:57]Does this whole book assume that we are working over the complex numbers?

[02:40:02]Ah, does it?

[02:40:04]I don't know if it does.

[02:40:31]Huh.

[02:40:34]I guess I'm interesting. Yeah. I look forward to learning what you all are talking about about this stuff. Shimma.

[02:40:44]I've heard of that.

[02:41:05]Yeah, y'all need exercises. Exercises are the way to to learn math. But you all know you all know what you're talking about. You all you all know.

[02:41:17]Hey, how's it going, Vob Edits? Good to see you. Good to see you.

[02:41:30]Yeah, I don't like hopping between books for a single for a single subject. like I need to just stick with one book for a while and one teacher's presentation of that material of that subject for a while and then once I feel like I've gotten a good amount of understanding then I can start comparing treatments.

[02:41:55]Um let me let me get the sales pitch up.

[02:42:04]Y'all, you can share this playlist with people. If anybody wants to, if you ever meet somebody in the grocery store and you're like, "Hey, you want to develop math as a hobby? I've been hanging out with this nerd on the internet. Uh, go to math can be a hobby.com and that will point people to the playlist."

[02:42:27]Um, let me catch up.

[02:42:46]Hey, Christopher. Yeah, I'm still doing guitar a little less nowadays. Um, we've gotten a lot of momentum here with math and it's math is also right now my professional career goal. Like I want to become a math teacher. I want to teach children algebra 1 and try to gently guide them toward category theory and abstract algebra.

[02:43:13]You know, I want to teach the little children representation theory, you know, just sneak it in there. Uh try to indoctrinate them early on to pure mathematics and proofbased mathematics. So yeah, I am uh I'm practicing a lot of math nowadays and we're we're finding, you know, some success over here. But I'm still doing the guitar stream, just not at this time of the day quite as much. Um, thank you.

[02:44:00]Oh, thank you. All right. Thanks, Arod.

[02:44:02]All vector spaces finite dimensional over the field of complex numbers. I see. Okay. Thank you all. Appreciate it.

[02:44:13]I forgot that that is stated exa explicitly.

[02:44:20]Okay, so I was extremely close. I will I will fix that argument. Um let's let's keep going though. Uh cuz Oh, damn.

[02:44:29]We're almost at three hours.

[02:44:34]Okay. Well, dude, I think that this is a good amount for today. Okay, I'm feeling really good about where we got.

[02:44:44]Um, we got to the definition of a unitary representation.

[02:44:59]Uh, and then we tried this proposition. Next time I want to I want to continue working that proposition again and work the proof a little bit better. Um, so we stated these things.

[02:45:18]Next time I want to prove all of these starred things.

[02:45:31]Um, so here's my to-do list for next time, for tomorrow, and we'll see if we can make some progress.

[02:46:03]Thank you everybody for watching. Um, this has been a real pleasure.

[02:46:10]It's it's really nice to be catching up and relearning or not relearning but like learning this stuff live with an audience and you know yeah I'll I'll be streaming at about the same time tomorrow. Um, so we're streaming weekdays around 9:00 a.m.

[02:46:39]Eastern time New York City. Uh, that's 3. No, what?

[02:46:48]2 PM GMT.

[02:46:54]Um, and this whole week we're doing representation theory.

[02:47:00]Uh, join the Discord, share the playlist, and yeah, great to see you all. Thank you for being here. Math can be a hobby.

[02:47:19]com if you want to share it with anybody. And uh Discord purple against black is horrible.

[02:47:30]Horrible.

[02:47:34]Hang out with us in the Discord.

[02:47:38]I'm not always super active in there.

[02:47:41]like I try to be sometimes, but I I kind of spend all of my social energy here on stream and then I need to recharge and do my own thing, you know, during the rest of the day. Uh, but I I am trying to be active in there as well. So, come hang out with us in the Discord and you can you can also get notifications whenever I'm going live and things like that. Um, thanks for watching and keep an eye out for episode two coming soon.

[02:48:14]Oh, wait, no, sorry.

[02:48:18]Episode 4 soon.

[02:48:23]Tomorrow.

[02:48:25]I I hope it's out tomorrow. I I think it should be out tomorrow. So, keep an eye out for that. And next time that uh we continue, we'll be getting to the proof of Mashki's theorem. I think. Um, thanks for watching. Bye. Bye everyone. Bye.

[02:48:42]Bye.

[02:48:51]Oh, uh, I I should do this also.

[02:48:55]Special freaking thanks. Special thanks to all the supporters of the channel.

[02:49:01]people who were collaborating, people who were helping so much with uh you know just hanging out and being regulars, people who are supporters, uh people who are in the courses, pe you know channel members, um yeah, appreciate you all very much. Thank you.

[02:49:20]Thank you for helping me, you know, uh do what I do what I do. Thank you all.

[02:49:27]All right, bye bye.