Initially dismissed, representation theory is now central to much of mathematics
abstractions.nautil.us
abstractions.nautil.us
Here is a simpler explanation.
A group is a collection of elements with a multiplication such that multiplication is associative, there is an identity, and every element has an inverse. Multiplication is generally NOT commutative.
The classic example is the set of all permutations of a set using function composition as multiplication. "Multiplication" is associative because (f o (g o h))(x) and ((f o g) o h)(x) works out to be f(g(h(x))) for any x. The identity permutation is "leave everything where it was". The inverse of a permutation is simply "map everything in reverse".
They spend time talking about S3. That is just the permutations of 3 objects. (Which they visualized as the corners of a triangle.)
Another classic example is the set of invertible nxn matrices using multiplication as multiplication.
A representation is just a function from a group to some set of invertible matrices such that F(x * y) = F(x) * F(y). Note that a representation does NOT have to be one-to-one - you can always just map everything to the identity matrix.
OK, so representations exist. Why would we care? The answer is that each representation of a group shows something about the structure of that group. And says it in a language that we have a lot of tools to work with. Admittedly, the identity representation doesn't say much useful. But the others do. And representations have provided ways to reduce a lot of hard problems to easier ones that we have a better chance to solve.
There are a lot of examples, but the most widely known example of a hard problem that used this idea as part of the solution was Andrew Wiles' solution of Fermat's Last Theorem.
Because they're writing for laymen. I have a degree in math and work in a subfield of CS that relies on math a lot, so I have a lot of practice explaining concepts to some of the smartest people I know without a math background. The tradeoff for your concise expression is that it uses jargon that adds cognitive overhead for people unfamiliar with college-level or higher math, even if they've got a high quantitative intelligence. A non-exhaustive list that would trip up, say, my friend with a PhD in pathology:
1) associative and commutative are not commonly-know terms
2) "A group is a collection of elements 'with' multiplication" is underdetermined. What does it mean to be "with"? The notion that multiplication isn't commutative is unintuitive without understanding that the definition of multiplication is flexible here, defined based on the type of the element in the group.
3) Using o as a composition operator is something that a layman is guaranteed never to have come across
4) What does invertible mean? It's not trivially obvious to a layman that some matrices A have no matrix B such that AB = I.
Etc etc etc. Note that each of these gaps and confusing points (for laymen) compound; I'll occasionally read a paper in a field I'm not an expert in (like economics), and the being bogged down every other sentence by confusion is a severe impediment to understanding how these pieces are composed into something novel. Hell, even when I understand each piece, sometimes the most foreign concepts haven't marinated long enough for me to use them as building blocks yet.
Whats your complaint here? "I have a math degree, how dare nautilus target anyone but me with their articles?"
I've lost contact with many of these after finishing my university, but they are still pretty common terms around me.
Function composition is a pretty trivial concept, but I referred specifically to using ∘ to represent it; this symbol isn't IME as widely-understood to the layman as simply using f(g(x)) (by contrast with college-level and higher math, where it's very widely in use).
The gist of my point was that even the smartest quantitatively-inclined people who went to my private school in a tony part of California aren't likely to have retained familiarity with these concepts unless their college education and/or career involves theoretical math. This is leaving aside people not as smart, or less quantitatively-inclined; my friends who fall into this bucket probably didn't remember what commutativity and associativity were by the time they _entered_ college, let alone later in life. Arguably this latter group wouldn't be part of the target audience interested in reading an article about theoretical math at all, but my friends in the former group certainly would read publications like Nautilus on general mathematical topics.
However that was a long time ago and standards have changed. http://www.corestandards.org/Math/Content/HSN/VM/C/9/ indicates that matrix math is now considered high school material across the USA.
But you know about theory vs practice.
The wikipedia sentence definition makes more sense to me: "a "representation" means a homomorphism from the group to the automorphism group of an object"
This seems natural, since if you take _any_ algebraic structure (e.g. a ring), then its automorphism group is a group. Which is a reason why group theory is worth studying - since _any_ algebraic structure can be studied using group theory by considering its automorphism group. (Also, an automorphism group is just the set of symmetries, so group theory can be described as the study of symmetry).
No one who graduated college with a degree in English is going to know what "multiplication is associative, there is an identity, and every element has an inverse. Multiplication is generally NOT commutative." means, and that's the first sentence...
As for what that sentence means, it means that given any two elements we can "multiply" them. Multiplication is associative, meaning that x * (y * z) is always the same as (x * y) * z. There is an identity simply means that an element we can call 1 exists such that 1 * x is always x. There is a multiplicative inverse means that for every x there is some element we can call x^(-1) such that x * x^(-1) = 1. By not commutative I mean that x * y doesn't have to be the same as y * x. (An example of non-commutative is that if you swap the first two items in a list then the first and third you get a different result than if you swap the first and third then swap the first two items.)
But still, you have answered my actual question.
What that site attempts to do is, at length, build some intuition about a topic, so that it can talk about it. By contrast my explanation just reminds someone who understands the fundamentals of the topic of the concepts so that we can start talking. So someone like you has a chance to understand something. Someone like me has to work to keep track of where they are, only to say, "You're just talking about group homeomorphisms."
For it to be useful to a layperson, you'll need to provide concrete examples and more of an explanation of each concept. Or, if you want to help people who are visual learners, something like a triangle.
With pictures, from the article.
A lot of people don't realize that the concepts of abstract groups, vector spaces, linear transformations, etc. were not particularly obvious to people, and a lot of results were developed in specific contexts that were later realized to be more general. After the development of abstract algebra, these subjects morphed into something much closer to the standard algebra curriculum we see today.
Dig until you get tired or bored - because you could really keep digging forever.
If you ever feel lost, read the masters - Euler and Dirichlet are two favorites. Here is a great book: https://www.amazon.com/Euler-Master-Dolciani-Mathematical-Ex...
If you get tired of that, read about the greats. MacTutor has a great archive. I especially like their biographies: https://mathshistory.st-andrews.ac.uk/
The most important thing (only my opnion) is to not lose the sense of wonder at the beauty of the universe as seen mathematically.
I don't know what your particular situation is - are you in university or school and studying math or learning math as you juggle a career? Any more specific advice depends on your circumstances and your objectives, but I really hope I was able to help somewhat.