@johncarlosbaez@mathstodon.xyz
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johncarlosbaez

@johncarlosbaez@mathstodon.xyz

I'm a mathematical physicist who likes explaining stuff. Sometimes I work at the Topos Institute. Check out my blog! I'm also a member of the n-Category Cafรฉ, a group blog on math with an emphasis on category theory. I also have a YouTube channel, full of talks about math, physics and the future.

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dpiponi, to random
@dpiponi@mathstodon.xyz avatar

Looked up speed of snails on Google to see if my USPS package "moving through network" from San Francisco is literally going at a snail's pace. Looks like snails would have to be 3 times faster to beat my package.

johncarlosbaez,
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@dpiponi - when the fantasy of package delivery by drones fell through, they switched to snails.

johncarlosbaez, (edited ) to random
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The precise location of the boundary between the knowable and the unknowable is itself unknowable. But we ๐‘‘๐‘œ know some details about ๐‘คโ„Ž๐‘ฆ this is true, at least within mathematics. It's being studied rigorously in a branch of theoretical computer science called 'meta-complexity theory'.

For some reason it's hard to show that math problems are hard. In meta-complexity theory, people try to understand why.

For example, most of us believe P โ‰  NP: merely being able to ๐‘โ„Ž๐‘’๐‘๐‘˜ the answer to a problem efficiently doesn't imply you can ๐‘ ๐‘œ๐‘™๐‘ฃ๐‘’ it efficiently. It seems obvious. But despite a vast amount of work, nobody has been able to prove it!

And in one of the founding results of meta-complexity theory, Razborov and Rudich showed that if a certain attractive class of strategies for proving P โ‰  NP worked, then it would be possible to efficiently crack all codes! None of us think ๐‘กโ„Ž๐‘Ž๐‘ก'๐‘  possible. So their result shows there's a barrier to knowing P โ‰  NP.

I'm simplifying a lot of stuff here. But this is the basic idea: they proved that it's probably hard to prove that a bunch of seemingly hard problems are really hard.

But note the 'probably' here! Nobody has ๐‘๐‘Ÿ๐‘œ๐‘ฃ๐‘’๐‘‘ we can't efficiently crack all codes. And this too, seems very hard to prove.

So the boundary between the knowable and unknowable is itself shrouded in unknowability. But amazingly, we can prove theorems about it!

https://www.quantamagazine.org/complexity-theorys-50-year-journey-to-the-limits-of-knowledge-20230817/

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@davidsuculum - that's nice; now I want to follow the precise proof of this "paradox" and its assumptions. But beware: in English "any" means both "some" (โˆƒ) and "all" (โˆ€). From your description I thought the paradox was claiming

"If some truth can be known then it follows that every truth is in fact known"

which is crazy, but in fact it claims

"If every truth can be known than it follows that every truth is in fact known"

This seems interesting and perhaps reasonable, since I believe not every truth can be known.

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@franchesko - the article I linked to mentioned meta-meta-complexity:

.....

Given the truth table of a Boolean function, determine whether it has high or low circuit complexity. They dubbed this the minimum circuit size problem, or MCSP.

[....]

MCSP is a quintessential meta-complexity problem: a computational problem whose subject is not graph theory or another external topic, but complexity theory itself.

Kabanets knew that he and Cai werenโ€™t the first to consider the problem they had dubbed MCSP. Soviet mathematicians had studied a very similar problem beginning in the 1950s, in an early attempt to understand the intrinsic difficulty of different computational problems. Leonid Levin had wrestled with it while developing what would become the theory of NP-completeness in the late 1960s, but he couldnโ€™t prove it NP-complete, and he published his seminal paper without it.

After that, the problem attracted little attention for 30 years, until Kabanets and Cai noted its connection to the natural proofs barrier. Kabanets didnโ€™t expect to settle the question himself โ€” instead he wanted to explore why it had been so hard to prove that this seemingly hard problem about computational hardness was actually hard.

โ€œIt is, in a sense, meta-meta-complexity,โ€ said Rahul Santhanam, a complexity theorist at the University of Oxford.

But was it hardness all the way down, or was there at least some way to understand why researchers hadnโ€™t succeeded in proving that MCSP was NP-complete? Kabanets discovered that, yes, there was a reason.

https://www.quantamagazine.org/complexity-theorys-50-year-journey-to-the-limits-of-knowledge-20230817/

johncarlosbaez, (edited )
@johncarlosbaez@mathstodon.xyz avatar

@aadmaa - of course, the question about what we can know about nonmathematical questions is infinitely harder than for mathematical ones. It may also be more important. But having studied it as a youth, I eventually decided it was too intractable to be worth spending more time on. I like working on things where I can make progress.

The question of unknowability for math is by comparison so clear and self-contained that I find it fascinating how even here we sink into deep quicksand of a self-referential sort. But since math is so clear and self-contained, we can prove amazing theorems about this quicksand. For example we can sometimes rigorously ๐‘๐‘Ÿ๐‘œ๐‘ฃ๐‘’ that for certain propositions, if we are unable to prove them, we are also unable to prove that we are unable to prove them. And so on.

The recent rise of meta-complexity theory is showing that such questions are relevant to cryptography, the study of efficient algorithms, etc. That's pretty amazing!

https://simons.berkeley.edu/programs/Meta-Complexity2023

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@TruthSandwich - Alas, that doesn't parse because P โ‰  NP is a single task, while the terms P and NP make sense only for ๐‘“๐‘Ž๐‘š๐‘–๐‘™๐‘–๐‘’๐‘  of tasks, like multiplying numbers or factoring numbers into primes. There are ๐‘™๐‘œ๐‘ก๐‘  of numbers, and we say multiplying numbers is in P because the time it takes grows polynomially as the numbers get bigger.

I could go on, but this is what happens when you make a joke to a mathematician.

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@leemph - my go-to formalization of "knowability" in math (and only in math) is "provability". For example, we'd know P โ‰  NP if we could prove it. What I'm hinting at in my post is that P โ‰  NP is unprovable. But while Goedel and many subsequent logicians were able to prove many interesting statements are unprovable (starting from various standard axioms), so far people have only been able to show that certain approaches to proving P โ‰  NP (so-called "natural proofs") lead to dizzying consequences, like our ability to tell the difference between random numbers and pseudorandom numbers.

It's completely possible that P โ‰  NP is unprovable and it's also impossible to prove it's unprovable, and it's also impossible to prove this.

For example, we know that if Goldbach's conjecture is true but unprovable, it's also impossible to prove that it's unprovable. So there are cases where uknowability shrouds itself in unknowability.

So, anyway, this is where my thoughts are on this - not broader concepts of "knowing".

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@leemph - regarding 1, you're exactly right: in math most of us have accepted that "mathematical knowledge" is relative to a set of axioms and deduction rules. This started with the discovery of non-Euclidean geometry, where "truths" of Euclidean geometry turned out to be consequences of axioms that no longer hold if you switch to a different sort of geometry. Later Goedel showed that any sufficiently powerful consistent finitely axiomatizable theory could never prove or disprove all statements formulated in its own language: there's always some statement P such that we can add either P or not(P) to the axioms and get a new such set of axioms that's still consistent.

  1. In constructivist logic to prove something exists means that you can, at least in principle, exhibit an example. In classical logic there are cases where you can prove something exists but not exhibit an example. So your attitude to this question is closely allied to whether you prefer classical or constructivist logic. Most really smart mathematicians realize that this, too, is another case of the relativity in part 1. I.e., neither constructivism or classical logic is "really true": they are just alternative sets of axioms, and it's worth exploring both.

  2. "If we prove T then do we have a proof that it is provable?" In the logics I know, exhibiting an example of something counts as a proof that it exists. So yes, in both classical and constructivist logic, we can get from a proof of P to a proof that P is provable. Part 2 was about the more problematic converse.

"In which deductive system should that meta-proof be carried out?" There are many choices. This is another instance of the relativity in 1.

(1/2)

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@leemph wrote: "<<For example, we know that if Goldbach's conjecture is true but unprovable, it's also impossible to prove that it's unprovable. So there are cases where unknowability shrouds itself in unknowability.>>"

Sorry, this was a stupid sentence, and you were right to be confused. Here's what I was trying to say:

"For example, if Goldbach's conjecture is true but unprovable, it's also impossible to prove that it's unprovable. So there are cases where unknowability shrouds itself in unknowability."

And normally I avoid using the word "true" in this context, since it doesn't really mean much to say a mathematical statement is "true" except as a shorthand for it being provable. If I were trying to be precise, I would have said this:

"For example, if neither Goldbach's conjecture nor its negation is provable, it's also impossible to prove that either of those is unprovable. So there are cases where unknowability shrouds itself in unknowability."

(2/2)

johncarlosbaez, to random
@johncarlosbaez@mathstodon.xyz avatar

Chemistry is like physics where the particles have personalities - and chemists love talking about the really nasty ones. It makes for fun reading, like Derek Lowe's column "Things I Won't Work With". For example, bromine compounds:

"Most any working chemist will immediately recognize bromine because we don't commonly encounter too many opaque red liquids with a fog of corrosive orange fumes above them in the container. Which is good."

And that's just plain bromine. Then we get compounds like bromine fluorine dioxide.

"You have now prepared the colorless solid bromine fluorine dioxide. What to do with it? Well, what you don't do is let it warm up too far past +10C, because it's almost certainly going to explode. Keep that phrase in mind, it's going to come in handy in this sort of work. Prof. Seppelt, as the first person with a reliable supply of the pure stuff, set forth to react it with a whole list of things and has produced a whole string of weird compounds with brow-furrowing crystal structures. I don't even know what to call these beasts."

https://www.science.org/content/blog-post/higher-states-bromine

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@ChateauErin - sounds fun, I will check out the YouTube video when pedaling at the gym. But I am glad that as a mathematician the things I work with can only demolish my brain from the inside.

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@gregeganSF - they should make up a vileness scale, like Moh's hardness scale.

TruthSandwich, to math
@TruthSandwich@fedi.truth-sandwich.com avatar

https://www.quantamagazine.org/strangely-curved-shapes-break-50-year-old-geometry-conjecture-20240514/

FYI, @johncarlosbaez

#math

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@TruthSandwich - nice! I hadn't seen that.

johncarlosbaez, (edited ) to random
@johncarlosbaez@mathstodon.xyz avatar

Fun article by John Psmith featuring some ferociously competitive mathematicians and physicists. A quote:

.....

In the 1696 edition of Acta Eruditorum, Johann Bernoulli threw down the gauntlet:

"I, Johann Bernoulli, address the most brilliant mathematicians in the world. Nothing is more attractive to intelligent people than an honest, challenging problem, whose possible solution will bestow fame and remain as a lasting monument. Following the example set by Pascal, Fermat, etc., I hope to gain the gratitude of the whole scientific community by placing before the finest mathematicians of our time a problem which will test their methods and the strength of their intellect. If someone communicates to me the solution of the proposed problem, I shall publicly declare him worthy of praise.

Given two points A and B in a vertical plane,
what is the curve traced out by a point acted on only by gravity,
which starts at A and reaches B in the shortest time."

This became known as the brachistochrone problem, and it occupied the best minds of Europe for, well, for less time than Johann Bernoulli hoped. The legend goes that he issued that pompous challenge I quoted above, and shortly afterward discovered that his own solution to the problem was incorrect. Worse, in short order he received five copies of the actually correct solution to the problem, supposedly all on the same day. The responses came from Newton, Leibniz, lโ€™Hรดpital, Tschirnhaus, and worst of all, his own brother Jakob Bernoulli, who had upstaged him yet again.

(1/2) (The fun part about Newton comes in part 2.)

https://www.thepsmiths.com/p/review-the-variational-principles

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@dougmerritt - I definitely think about it! I don't tend to think of Fermat as an otherworldly genius, I think of him as a very smart lawyer who corresponded with lots of mathematicians and physicists. But looking at his biography, I realize I may be underestimating him. Newton was an otherworldly genius of an articulate and versatile sort: the sort who changes the course of civilization. Ramanujan seems like an inarticulate, specialized genius with a completely incomprehensible intuitive ability to spot fascinating formulas - perhaps a mere curiosity in the grand scheme of things, but extremely puzzling.

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@pieter - Then I won't read that.

johncarlosbaez, (edited ) to random
@johncarlosbaez@mathstodon.xyz avatar

There's a dot product and cross product of vectors in 3 dimensions. But there's also a dot product and cross product in 7 dimensions obeying a lot of the same identities! There's nothing really like this in other dimensions.

We can get the dot and cross product in 3 dimensions by taking the imaginary quaternions and defining

vโ‹…w= -ยฝ(vw + wv), vร—w = ยฝ(vw - wv)

We can get the dot and cross product in 7 dimensions using the same formulas, but starting with the imaginary octonions.

The following stuff is pretty well-known: the group of linear transformations of โ„ยณ preserving the dot and cross product is called the 3d rotation group, SO(3). We say SO(3) has an 'irreducible representation' on โ„ยณ because there's no linear subspace of โ„ยณ that's mapped to itself by every transformation in SO(3).

Much to my surprise, it seems that SO(3) also has an irreducible representation on โ„โท where every transformation preserves the dot product and cross product in 7 dimensions!

It's not news that SO(3) has an irreducible representation on โ„โท. In physics we call โ„ยณ the spin-1 representation of SO(3), or at least a real form thereof, while โ„โท is called the spin-3 representation. It's also not news that the spin-3 representation of SO(3) on โ„โท preserves the dot product. But I didn't know it also preserves the cross product on โ„โท, which is a much more exotic thing!

In fact I still don't know it for sure. But @pschwahn asked me a question that led me to guess it's true:

https://mathstodon.xyz/@pschwahn/112435119959135052

and I think I almost see a proof, which I outlined after a long conversation on other things.

The octonions keep surprising me.

https://en.wikipedia.org/wiki/Seven-dimensional_cross_product

johncarlosbaez, (edited )
@johncarlosbaez@mathstodon.xyz avatar

@dougmerritt - the vector cross product in 7 dimensions is a very mysterious thing, but luckily it's just a spinoff of the octonions: to the extent we understand the octonions we understand it. I've been thinking about it for a long time. I thought I understood it pretty well.

The thing that's interesting me now - the thing I learned just yesterday! - is that just as 3d rotations act on 3d vectors in a way that preserves their dot product and cross product, 3d rotations also act on 7d vectors in a way that preserves their dot product and cross product. That's bizarre.

The "irreducibility" business is a way of saying that we're not getting this to happen using a cheap trick. If we dropped the irreducibility condition, we could think of 3d rotations as 7d rotations that just happen to only mess around with 3 of the coordinates. We are not doing that here!

So this is weird. By the way, in general, 7d rotations DON'T act on 7d vectors in a way that preserves their dot product and cross product. Only certain special ones do.

(It takes 28 numbers to specify a general rotation in 7 dimensions, and they all preserved the dot product of 7d vectors. Far fewer preserve the cross product too: those can be specified using only 14 numbers.)

johncarlosbaez, (edited )
@johncarlosbaez@mathstodon.xyz avatar

@internic - The cross product is not associative! But you got the idea. Let me spell it out in painful detail. A "vector product algebra" is a vector space with an inner product I'll call the dot product and denote by

โ‹…: Vยฒ โ†’ โ„

and also a bilinear antisymmetric operation called the cross product

ร— : Vยฒ โ†’ V

obeying

u โ‹… (v ร— w) = v โ‹… (w ร— u)

and

(u ร— v) ร— u = (u โ‹… u) v - (u โ‹… v) u

These are what I meant by "the usual identities". They imply a bunch more identities.

There exist only four vector product algebras! The only interesting ones are the 3d and 7d ones, since in the 0d and 1d examples the cross product is zero.

For more, try:

https://www.math.uni-bielefeld.de/~rost/data/vpg.pdf

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@dmm - don't be shy about asking questions, and don't be scared to ask 'elementary' questions. I really like explaining math. Whatever might help you, I'm willing to give it a try.

johncarlosbaez, (edited )
@johncarlosbaez@mathstodon.xyz avatar

@pschwahn - In our previous conversation I conjectured a representation of SO(3) on the imaginary octonions, and by now I've checked it really is a representation. Once we can know it's irreducible, we know it's the spin-3 irrep.

To get this rep we start by picking a basic triple of imaginary octonions, say i, j, โ„“. They span a 3d space. Any rotation of this 3d space rotates our basic triple to some other basic triple. This gives an automorphism of the octonions and thus a transformation of the space of imaginary octonions. So we get a rep of SO(3) on the 7d space of imaginary octonions! That's all there is to it.

We could either prove this is irreducible, or check by computation that it's the spin-3 rep. I can explain more explicitly how rotating our basic triple defines an automorphism of the octonions. Showing this gives the spin-3 rep may be easier at the Lie algebra level.

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@internic @ppscrv - it's painful to type.

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@pschwahn - Oh my god, how dumb I am! span{i,j,โ„“} is definitely invariant because I'm rotating those vectors into linear combinations of themselves. So... back to the drawing board.

By the way, it was not completely trivial to check that if we rotate i, j, โ„“ we get a new basic triple, because the concept of basic triple is defined using octonion multiplication. So when I succeeded, I thought I was being clever. I seem to be getting a bunch of SO(3) subgroups of Gโ‚‚ (each basic triple gives me one), but none that act irreducibly on the imaginary octonions.

I suspect that a 'generic' SO(3) subgroup of Gโ‚‚ will act irreducibly. Sometimes 'generic' things are the ones for which there's no simple formula. But I still hope there's something nice going on here. I want to connect 3d geometry to 7d or 8d geometry in a nice way.

There's a nice way to build the octonions from โ„ยณ where you take the exterior algebra ฮ›(โ„ยณ), which is 8-dimensional, and give it a multiplication. SO(3) acts on ฮ›(โ„ยณ) in the usual way, and this preserves the octonion multiplication, but unfortunately it acts reducibly since it preserves each grade ฮ›โฑ(โ„ยณ).

So we need something more gnarly.

monsoon0, to random
@monsoon0@mathstodon.xyz avatar

A proof is an amazing wonderful ๐ŸŽ‰ thingโ€ฆ So I am wondering why the word โ€œonlyโ€ is in this sentence: โ€œResearchers have obtained only mathematical proofs that quantum computers will offer large gains over current, classical computersโ€ https://www.nature.com/articles/d41586-023-01692-9

johncarlosbaez, (edited )
@johncarlosbaez@mathstodon.xyz avatar

@monsoon0 - yes, the math of quantum computing is very helpful. I believe the article was trying to say, in a clumsy way, that the math of quantum algorithms is far ahead of our technology for implementing these algorithms, and there's no guarantee that we'll succeed in implementing them.

As for @drdunk's statement, it's a fact that nobody has proved really good lower bounds on the asymptotic complexity of classical computations For this reason, nobody can prove that quantum algorithms are definitively faster, asymptotically, than classical ones could ever be. What people do instead is prove that some quantum algorithms are faster, asymptotically, than the current ๐‘๐‘’๐‘ ๐‘ก ๐‘˜๐‘›๐‘œ๐‘ค๐‘› classical algorithms.

Consider Shor's algorithm for prime factorization:

https://en.wikipedia.org/wiki/Shor%27s_algorithm

This is one of the main examples of a quantum algorithm that seems to beat what we can do classically. The Wikipedia article states what's known about it. Shor's algorithm for factoring an integer N takes time that's bounded by a polynomial in log N (Wikipedia states a very precise bound that's below O((log N)ยณ)). The current best known classical algorithm takes time that grows faster than any polynomial in log N. But nobody has proved that it's impossible to use classical algorithms to factor integers in a time that's bounded by a polynomial in log N. It's widely believed to be true, but not proved.

To my mind, the most interesting thing about this is the question of why it's so hard to prove good lower bounds on the complexity of algorithms! There's something very deep about this, which "meta-complexity" theory is trying to tackle:

https://www.quantamagazine.org/complexity-theorys-50-year-journey-to-the-limits-of-knowledge-20230817/

pschwahn, to random German
@pschwahn@mathstodon.xyz avatar

Non-semisimple Lie groups are so weird. Weyl's unitarian trick does not work for them. So I need to constantly remind myself that:

  1. representations of GL(n,โ„‚) are not determined by their character,
  2. not every finite-dimensional representation of GL(n,โ„‚) is completely reducible,
  3. Finite-dimensional GL(n,โ„‚)-representations are not in 1:1-correspondence with finite-dimensional U(n)-representations.

However these work when you look only at irreducible representations, or when you replace GL by SL (and U by SU). The archetypical counterexample is given by the (reducible but indecomposable) representation
[\rho: \mathrm{GL}(1,\mathbb{C})=\mathbb{C}^\times\to\mathrm{GL}(2,\mathbb{C}):\quad z\mapsto\begin{pmatrix}1&\log |z|\0&1\end{pmatrix}.]
(Example shamelessly stolen from: https://math.stackexchange.com/questions/2392313/irreducible-finite-dimensional-complex-representation-of-gl-2-bbb-c)

Turns out that entire StackExchange threads can be wrong about this (for example https://math.stackexchange.com/questions/221543/why-is-every-representation-of-textrmgl-n-bbbc-completely-determined-by), so be wary!

johncarlosbaez, (edited )
@johncarlosbaez@mathstodon.xyz avatar

@pschwahn @AxelBoldt

"However for a group such as GL(๐‘›,โ„) it looks like the issue persists - a representation such as ๐‘”โ†ฆ|det(๐‘”)|สท for non-integer w will be real-analytic, but it will not correspond to a representation of U(n)."

U(n) is not a maximal compact subgroup of GL(n,โ„) - it's not even contained in GL(n,โ„). So there's no way to restrict a representation of GL(n,โ„) to U(n), and you shouldn't expect an equivalence (or even a functor) from the category of representations of GL(n,โ„) to those of U(n).

The maximal compact subgroup of GL(n,โ„) is O(n), so you can restrict representations of GL(n,โ„) is O(n). But a bunch of different real-analytic representations of GL(n,โ„) restrict to the same representation of O(n), like all the representations ๐‘”โ†ฆ|det(๐‘”)|สท. If I remember correctly this particular example is the "only problem". Of course it has spinoffs: you can tensor any representation of GL(n,โ„) by a representation ๐‘”โ†ฆ|det(๐‘”)|สท and get a new one which is the same on O(n).

I hope I'm remembering this correctly: every finite-dimensional smooth representation of GL(n,โ„) is completely reducible, and every irreducible smooth representation comes from one described by a by Young diagram, possibly tensored by a representation ๐‘”โ†ฆdet(๐‘”)สท where w is some real number, possibly also tensored by a representation ๐‘”โ†ฆ|det(๐‘”)|สท where w is some real number.

It's a lot easier to find treatments of the 'algebraic' representations of GL(n,โ„), and it's even easier to find them for SL(n,โ„).

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@pschwahn @AxelBoldt - Ugh! I should have stuck with rational representations, which is what people usually talk about when studying representations of linear algebraic groups.

I'm pretty sure that every finite-dimensional rational representation of GL(n,โ„) is completely reducible, and every irreducible rational representation comes from one described by a by Young diagram, possibly tensored by a representation ๐‘”โ†ฆdet(๐‘”)โฟ where n is some integer.

(There is some overlap here since the nth exterior power of the tautologous representation, described by a Young diagram, is also the representation ๐‘”โ†ฆdet(๐‘”).)

It's annoying that the basic facts about finite-dimensional representations of GL(n,โ„) aren't on Wikipedia! Someday I'll have to put them on there... once I get enough references to make sure I'm not screwing up!

johncarlosbaez, (edited )
@johncarlosbaez@mathstodon.xyz avatar

@pschwahn @AxelBoldt - right, that's one way to proceed. I've been doing a lot of work lately with representations of GL(n,๐”ฝ) for ๐”ฝ an arbitrary field of characteristic zero. For subfields of โ„‚ this trick of complexifying and reducing to the case ๐”ฝ = โ„‚ works fine. But in fact the representation theory works exactly the same way even for fields of characteristic zero that aren't subfields of โ„‚!

It's not that I really care about such fields. I just find it esthetically annoying to work only with subfields of โ„‚ when dealing with something that's purely algebraic and shouldn't really involve the complex numbers. So I had to learn a bit about how we can develop the representation theory of GL(n,๐”ฝ) for an arbitrary field of characteristic zero. Milne's book ๐ด๐‘™๐‘”๐‘’๐‘๐‘Ÿ๐‘Ž๐‘–๐‘ ๐บ๐‘Ÿ๐‘œ๐‘ข๐‘๐‘  does this, and a preliminary version is free:

https://www.jmilne.org/math/CourseNotes/iAG200.pdf

but unfortunately it's quite elaborate if all you want is the basics of the representation theory of GL(n,๐”ฝ).

(For ๐”ฝ not of characteristic zero everything changes dramatically, since you can't symmetrize by dividing by n!. Nobody even knows all the irreps of the symmetric groups.)

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