Initially dismissed, representation theory is now central to much of mathematics
I have no idea how nautil manages to so consistently talk about mathematical concepts in a way that is so hard to make heads or tails of.<p>Here is a simpler explanation.<p>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.<p>The classic example is the set of all permutations of a set using function composition as multiplication.  &quot;Multiplication&quot; 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 &quot;leave everything where it was&quot;.  The inverse of a permutation is simply &quot;map everything in reverse&quot;.<p>They spend time talking about S3.  That is just the permutations of 3 objects.  (Which they visualized as the corners of a triangle.)<p>Another classic example is the set of invertible nxn matrices using multiplication as multiplication.<p>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.<p>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&#x27;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.<p>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&#x27; solution of Fermat&#x27;s Last Theorem.