
A ‘Useless’ Perspective That Transformed Mathematics - exanimo_sai
https://www.quantamagazine.org/the-useless-perspective-that-transformed-mathematics-20200609/
======
biddlesby
Ever since my undergraduate I've wanted to understand what representation
theory is and the rough outline of how Andrew Wile's proof was constructed.

This article gave me both in a very understandable and engaging way. Thank you
to the author!!!

~~~
sdenton4
A somehow-much-less-popular way to come at representation theory is via
generalized Fourier transforms. The irreducible representations are basically
different "frequencies" which you can use to decompose functions on a group.
And in fact, all of the major Fourier theorems carry over perfectly... There's
even analogues of the fast Fourier transform, via chains of subgroups.

~~~
memexy
Are there any good references for this approach (via generalized Fourier
transforms)?

~~~
bollu
There is a thesis by Kondor: "Group theoretical methods for machine learning"
\-
[https://people.cs.uchicago.edu/~risi/papers/KondorThesis.pdf](https://people.cs.uchicago.edu/~risi/papers/KondorThesis.pdf)
which provides links to the associated literature.

The documentation of his library (SnOB) is great:
[http://www.gatsby.ucl.ac.uk/~risi/SnOB/SnOB/SnOB.pdf](http://www.gatsby.ucl.ac.uk/~risi/SnOB/SnOB/SnOB.pdf)

introductory slides on representation theory in machine learning:
[http://www.gatsby.ucl.ac.uk/~risi/courses/mini08/symmetric.p...](http://www.gatsby.ucl.ac.uk/~risi/courses/mini08/symmetric.pdf)

Full mini course:
[http://www.gatsby.ucl.ac.uk/~risi/courses/mini08/mini08.html](http://www.gatsby.ucl.ac.uk/~risi/courses/mini08/mini08.html)

~~~
memexy
Thanks. These all look very readable.

------
IngoBlechschmid
Representation theory, the subject of the submitted article, is also at the
core of our understanding of the particle zoo in physics. A tutorial aimed at
mathematicians is here:

John Baez, John Huerta. The Algebra of Grand Unified Theories.
[https://arxiv.org/abs/0904.1556](https://arxiv.org/abs/0904.1556)

~~~
benrbray
Great link, thanks! I alway enjoy reading summaries of other fields written
by/for mathematicians, since the assumption of mathematical background allows
a lot of fluff to be cut out (compared to, e.g. reading a textbook about
physics intended for physics students).

For background reading, the notes by Teleman 2005, "Representation Theory" [1]
are a good intro to the topic.

Learning about representation theory helped me understand the power of
thinking about mathematical objects in terms of their action on other objects.
Just as representation theory studies group actions on vector spaces, the
theory of modules is best described as the study of ring actions on
commutative groups. Many things happen to be rings (e.g. endomorphism rings of
functions, where addition=pointwise addition and multiplication=composition),
and modules allow us to apply (almost all of) vector space theory to better
understand ring-like objects.

[1]
[https://math.berkeley.edu/~teleman/math/RepThry.pdf](https://math.berkeley.edu/~teleman/math/RepThry.pdf)

[2] Another favorite from John Baez:
[https://groups.google.com/d/msg/sci.physics.research/aiMUJrO...](https://groups.google.com/d/msg/sci.physics.research/aiMUJrOjE8A/jGy2N3IaajwJ)

------
ur-whale
Very good article in that it taught me something I didn't know existed, and
explained it very well.

However, I wish the article gave an example on how to actually construct a
mapping between e.g. a small finite group and the _actual_ matrices in the
representation, to - sort of - get a feel for why that actually works.

I suspect there must be some sort of canonical method/algorithm to get from
the "multiplication table" of a finite group to each matrix in a
representation, but I haven't been able to find a reference.

Would anyone have pointers?

Also, jumping from the group to the character table (which seem to imply that
there is indeed an algorithm to compute all possible representations) without
having been told _how_ the mapping is constructed feels like a rather big
mental jump (and what makes the trace of the matrices important, btw - rather
than, say, the determinant?).

~~~
eigenket
Given the multiplication table of a (finite) group there are a couple of
"obvious" representations you can pick. Firstly theres always the trivial
representation that just sends everything to the identity.

More interestingly you can use the group elements as labels for an orthonormal
basis for a vector space (e.g. your orthonormal set is {v_g for all g in G}).
Then have the representation act by the usual group operation on the labels
R_h[v_g] = v_{hg}. Then once you have an operation defined on a basis
extending it linearly to the rest of the vector space is easy. This is (left)
_regular representation_.

[https://en.wikipedia.org/wiki/Regular_representation](https://en.wikipedia.org/wiki/Regular_representation)

Edit:

Intuition for why the trace is important comes from looking at permutation
matrices - the trace is the sum of the diagonal elements of the matrix and the
number of 1s on the diagonal is exactly the number of points that are
invariant under the permutation - e.g. the permutation

[[1,0,0,0], [0,1,0,0], [0,0,0,1], [0,0,1,0]]

has trace 2, and the permutation leaves the first two elements unchanged and
flips the last two.

~~~
JadeNB
Two remarks on this nice example:

(1) The word 'orthonormal' doesn't have to be there; representations are _a
priori_ just linear actions; if they carry an invariant inner product, that is
a bonus ('unitary representation'). For compact (including finite) groups, the
invariant inner product is no extra information, because we could pick any
(not necessarily invariant) inner product and average it with respect to Haar
measure to get an invariant one. This product is unique up to a scalar for an
_irreducible_ representation, but not in general (and the left-regular
representation is _never_ irreducible for a non-trivial finite group).

(2) For people who are more used to cycle notation than the matrix
representation of permutations, your permutation is (3 4). In roster notation,
it's

1 2 3 4

1 2 4 3

------
chmaynard
Peter Woit posted some comments:

[https://www.math.columbia.edu/~woit/wordpress/?p=11776](https://www.math.columbia.edu/~woit/wordpress/?p=11776)

------
memexy
This is a really good article on group theory and their representations but
the article title is unfortunate. It could have just been "Representation
theory and how it transformed mathematics".

~~~
klyrs
Perhaps The Unreasonable Effectiveness of Representation Theory

~~~
DC-3
Don't we have enough of these memetic titles? It's hard to go a day without 'X
for fun and profit' or 'Y considered harmful'.

~~~
memexy
He's referencing "The Unreasonable Effectiveness of Mathematics in the Natural
Sciences" by the physicist Eugene Wigner
([https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness...](https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences)).

> In the paper, Wigner observed that the mathematical structure of a physical
> theory often points the way to further advances in that theory and even to
> empirical predictions.

I think it's a good analogy since it applies exactly how Wigner meant it
except in this case it is a branch of mathematics being applied to mathematics
instead of physics. Representation theory has been "unreasonably useful and
effective" which is what the original article was about.

~~~
klyrs
Hi, I'm a woman. We still exist in tech spaces. Please use "they" when you
don't know somebody's gender, or "she" for me now that you know. Thanks

~~~
memexy
Noted. But I don't pay much attention to user names so if we run into each
other again I'm likely to make the same mistake.

~~~
klyrs
I don't care if you remember me. Remember this.

> Please use "they" when you don't know somebody's gender

------
emmelaich
Geordie Williamson has a fascinating history. He's the youngest living Fellow
of the Royal Society.

[https://en.wikipedia.org/wiki/Geordie_Williamson](https://en.wikipedia.org/wiki/Geordie_Williamson)

------
poletopole
Sometimes I go deep wiki some nights in mathematics. Representation theory and
Set Theory are the only sane ways I know of on how to approach higher
mathematics. But it’s their practical applications in software that amazes me.

Take Young Tabs
([https://en.wikipedia.org/wiki/Young_tableau](https://en.wikipedia.org/wiki/Young_tableau))
and Hook Lengths for example. I’ve been playing with the concept that you
could use Young Tabs and Hook Lengths to represent groups of FSMs in metric
space if you wanted to know mathematically if one FSM could be topologically
sorted into a congruent FSM.

~~~
ssivark
Fascinating! Any references elaborating on this? (By FSMs, I suppose you mean
Finite State Machines?)

------
sva_
>“Mathematicians basically know everything there is to know about matrices.
It’s one of the few subjects of math that’s thoroughly well understood,” said
Jared Weinstein of Boston University.

I'm genuinely curious how such a statement can be made. I've recently been
wondering about if it were possible to prove that one's theorems are
exhaustive about a 'space' of possible theorems in an axiomatic system (or
some subset thereof, since I'd assume such a space might be infinite
(perhaps)).

How can we know that there aren't some really surprising properties of
matrices that we've previously been unaware of? As far as I can see, we can
merely make a statement about that it fits well together with other (limited)
findings we've made so far?

~~~
nvusuvu
Just this year I read that a new insight about eigenvalues/eigenvectors had
just been discovered, shocking the math world that a new insight from linear
algebra could be discovered.

~~~
andrepd
Can you share it?

~~~
jwmerrill
I bet it was “Eigenvectors from Eigenvalues”

* [https://www.quantamagazine.org/neutrinos-lead-to-unexpected-...](https://www.quantamagazine.org/neutrinos-lead-to-unexpected-discovery-in-basic-math-20191113/)

* [https://terrytao.wordpress.com/2019/08/13/eigenvectors-from-...](https://terrytao.wordpress.com/2019/08/13/eigenvectors-from-eigenvalues/)

* [https://arxiv.org/abs/1908.03795](https://arxiv.org/abs/1908.03795)

------
lawrenceyan
Are there any textbooks that provide a good introduction into Representation
Theory, assuming a decent level of undergraduate mathematics as a foundation?

~~~
benrbray
I recently did a deep dive into the topic, so here's a list of things I came
across that helped me:

[1] Teleman 2005:
[https://math.berkeley.edu/~teleman/math/RepThry.pdf](https://math.berkeley.edu/~teleman/math/RepThry.pdf)

[2] Khonvanov List of Repr Theory Resources
[http://www.math.columbia.edu/~khovanov/resources/](http://www.math.columbia.edu/~khovanov/resources/)

[3] Huang 2010, "Fourier-Theorietic Probabalistic Inference over Permutations"

[4] Woit, Topics in Representation Theory Course,
[http://www.math.columbia.edu/~woit/repthy.html](http://www.math.columbia.edu/~woit/repthy.html)

[5] Gallier 2013, "Spherical Harmonics and Linear Representations of Lie
Groups"
[https://www.cis.upenn.edu/~cis610/sharmonics.pdf](https://www.cis.upenn.edu/~cis610/sharmonics.pdf)

~~~
lawrenceyan
Thank you!

------
krackers
See also the newer post:
[https://news.ycombinator.com/item?id=23549897](https://news.ycombinator.com/item?id=23549897)
which has another presentation of this material.

------
nickray
Anybody have an actual quote of Burnside's doubt? Article etc.

~~~
CrazyStat
I believe the Burnside quote comes from the preface of his 1897 book:

> It may then be asked why, in a book which professes to leave all
> applications on one side, a considerable space is devoted to substitution
> groups; while other particular modes of representation, such as groups of
> linear transformations, are not even referred to. My answer to this question
> is that while, in the present state of our knowledge, many results in the
> pure theory are arrived at most readily by dealing with properties of
> substitution groups, it would be difficult to find a result that could be
> most directly obtained by the consideration of groups of linear
> transformations.

