A WEIRD VECTOR SPACE: Building a Vector Space with Symmetry | Nathan Dalaklis

[@CHALKND]

I obviously struggled with my camera here... At least the Chalkboard looks good! 😂😅

[@sadface7457]

Not even 7minute in and the notion that group symmetry are permutation and these permutation can be transposed on graph is fascinating.

[@CHALKND]

😀 Although I didn’t really go too far deep into the group theory stuff, I’m glad it was recognizable! Thanks!

[@sadface7457]

@@CHALKND The video would have be better surved as trilogy of videos to give each concept room to breath, but excellent non-the-less.

[@CHALKND]

Haha I agree! 😂 When I was drafting the boards for this the thought definitely crossed my mind, but then I realized that I hadn’t published in a few weeks 🙃

[@sadface7457]

@@CHALKND I understand that a polynomial is vector space. Is symmetric polynomial just polynomial where the invatient (doesn't chanhe) if you change the variables p(x, y) = x + xy+ y is symmetric because x can become y and vice versa without changing anything. My understanding is that polynomials vectors because you can decompose then into the dot product of two vectors. The additional operation comes from it being a ring.

[@CHALKND]

You’re right that function is symmetric because you can switch x and y and still have the same function. Now thinking about the dot product decomposition; Are we talking about taking the coefficients to get a vector in R^n and doing the dot product on those? Sure that works for polynomials in R[x] that are restricted to be at most degree n, but here I think you would need to do a bit more work since we have more than one variable we can’t just translate x^n to the nth position in the vector and we’re not setting a minimal degree for these symmetric polynomials. Also, vectors do not require a dot product to be vectors. In the definition we only require that they ‘play well’ with the operations of + and scalar multiplication. There are many different inner products that one could define on any given vector space, it is a different mathematical object that one can add to a vector space, although in many cases helpful, it is not required.

[@samuelmat97]

I like your videos and the way that you present. Congrats ! About the content, I've never seen it before. Interesting

[@CHALKND]

Thanks! I’m glad I was able to show you something that you haven’t seen before!

[@hannahmiller5515]

I can't follow what you're saying at all, but I've listened to this video twice while at work because your voice is very reassuring haha!! I hope to understand what you're talking about some day

[@nicholascimaszewski4453]

Love this video. I had a question though - I thought that function composition notation would necessitate that you write "f circle pi" rather than "pi circle f", since the rightward function gets applied first, and because pi is a permutation, it should be applied to (x1,...,xn), rather than the scalar output of f(x1,...,xn), right?

[@CHALKND]

Hi Nicholas, this is a great question! In the context of this video, we are thinking of π as a function from a polynomial ring R[v_1,v_2,v_3,v_4,..., v_n] to itself. So in this context, the output of f is not a scalar. Instead f constructs an abstract polynomial that lives in the polynomial ring R[v_1,v_2,v_3,v_4,..., v_n], and that's why the parenthetical statement "(π operates on indices)" around 11:03 still makes sense with the notation of π \circ f. With this in mind, the vector space of symmetric polynomials of the polynomial ring R[v_1,v_2,v_3,v_4,..., v_n] is the subset of R[v_1,v_2,v_3,v_4,..., v_n], call it S, where every permutation of indices π just acts like the identity map when restricted to S. If you're taking the approach that f is a function from R^n -> R for example, then yes, I agree, the notation would need to be flipped.

[@nicholascimaszewski4453]

@@CHALKND Got it, thanks. Love your channel, excited to see more videos!

[@iangrant8174]

Cool, because this whole "world is on fire thing" is _boring!_ 😀💓

[@CHALKND]

😂😂😂

[@iangrant8174]

This is a really beautiful little thing!!! Here are some vague ideas for applications: logicafterthought.blogspot.com/2020/04/how-to-be-genius-part-ii.html

[@iangrant8174]

And it does sound _very weird,_ but I love _anything_ to do with cemeteries!

[@iangrant8174]

This is awfully vague, but I have a feeling that the early history of vector spaces included a (somewhat prematurely truncated) branch of development via Hamilton's ideas of quaternions, and Clifford Algebras which may turn out to be very interesting. See arxiv.org/abs/0907.5356

[@CHALKND]

This looks like an interesting read. I'll definitely take a look! Thanks! 😀