Symmetry, Algebra, and the Monster
I really do like Quanta Magazine but it sometimes bogs down in background for 75% of the article hardly saying anything about the subject in hand. Anything vaguely concerning groups-physics seems to come with the same lukewarm re-review of symmetry. They ought to hit that once and or for all then give a link to it spending the majority of their time on the subject at hand. I&#x27;m also a bit jealous of the writers here: they seems to have fun job and I&#x27;d like to interview the same guys&#x2F;gals and I&#x27;d love to type something up too.<p>The Langlands Programme was an effort in group theory to develop a representation (a way to generate all the elements in a group as far as I understand it) for all possible groups. Groups depend both on the operation and the elements it operates over. Groups satisfy the four definitional requirements. Assume addition here:<p>- If &#x27;a&#x27; is an element in the group there must be another element &#x27;e&#x27; such that a+e=a. e.g. 2+0=2<p>- If &#x27;a&#x27; is in the group there must be another element in the group &#x27;b&#x27; such that a+b=e e.g. 2 + (-2) = 0. This is the inverse requirement.<p>- The group operation must be associative a+(b+c) = (a+b)+c<p>- If &#x27;a,b&#x27; is in the group then c=a+b and &#x27;c&#x27; must be in the group. This is called closure<p>- If, optionally, a+b = b+c then the group is called Abelian. Notably when a,b are matrices and instead of addition we do multiplication a<i>b &lt;&gt; b</i>a generally. So being Abelian is hardly automatic.<p>If you try to make a group on the addition operator using only whole numbers you&#x27;ll soon see you&#x27;ll break the inverse requirement e.g. for some whole number &#x27;a&#x27; there&#x27;s no other whole number &#x27;b&#x27; s.t. a+b=0. You&#x27;ll need the negative of the whole numbers too. Therefore grade school arithmetic addition is a group on (+,Z) where Z is all integers.<p>Note that the number of elements in Z is infinite. You&#x27;ll get the same sort of infinite sets trying addition over complex numbers, or reals, or fractions some being countably infinite and some being uncountably infinite.<p>Returning to the monster groups, how do we get a group with a finite number of elements in it? Well, there&#x27;s a pseudo way to do this: define addition (or your operation in general) to work over a modulus. So we could define Abelian addition over the three elements 0,1,2 like this:<p>0+0 = 0<p>0+1 = 1<p>0+2 = 2<p>1+0 = 0<p>1+1 = 2<p>1+2 = 0 we wrap mod 3<p>2+0 = 2<p>2+1 = 0<p>2+2 = 1 e.g. 2+1+1 = 0 + 1 = 1<p>Ok, what do we add to 1 to get 0 for inverse? What&#x27;s 1&#x27;s inverse? It&#x27;s 2 since 1+2=0 and 0 is the identity element. Other examples can be worked out from above. And because it&#x27;s Abelian we can reverse the order of the elements in the example and get the same thing.<p>Typically these kind of mod groups like this mod-3 example can be done mod-N where N is a prime. Granted, it&#x27;s a trick to make closure work without having one&#x27;s hand forced through in an infinite number of more elements.<p>Thus most of the possible groups are either of the (+,Z) form addition over Z, (.,R) multiplication over reals, (+,C) addition of complex numbers kind etc. And the second major swath are groups operating over a prime modulus.<p>Is that it? No! The Langlands programme found there are indeed other groups which have a finite number of elements but in a way that&#x27;s not contrived like the modulus example. To get a finite number of elements in a non-contrived way requires symmetry. You start with &#x27;e&#x27; and two other elements you like a,b. As you try a+e, b+e, a+b check to see if you get resultant elements in your group. If no, throw that in. And repeat.<p>Now, for a while all you do is get more and more elements until suddenly because of symmetry, you don&#x27;t. There&#x27;s something in the nature of the elements and the operations whereby they only transform themselves into unique elements up to some point then they don&#x27;t.<p>This is why the article plays up symmetry without really saying why. Imagine a+b means rotate a by b. Then you can do line symmetry of (a+b)+c through c and on and on an on and on ... until at some point you&#x27;ve played through all the symmetry on all the permutation and choices of elements until it goes back into something you started with earlier. Extending the analogy monster groups have rotation, translate, mirror, and zillion other kinds of symmetries ...<p>The monster groups thus are finite groups where symmetry is in play except that takes a huge number of elements -- the article mentions 10^53, a monster sized number --- at which point there&#x27;s no new uniqueness. Evidently then (Z+) lacks the symmetry of these monster groups since you need an infinite number of elements there.<p>I can&#x27;t answer in any detail how that carries over to physics except to repeat that physics people have seen the same math forms in monster groups in their theories. I also cannot answer what operation is in play in the monster groups --- well in most cases in algebra on these kinds of questions --- the group operation gets less focus. More focus goes on the <i>form of the elements</i>.<p>I don&#x27;t know what an element of the monster group looks like. It almost certainly is not a single number ... it&#x27;s a more complex thing like a matrix. I do know there&#x27;s some ~25 monster groups ... so this process of it gets bigger until it doesn&#x27;t happens in a few variations collectively called the sporadic groups. Sporadic in the sense there are but a few of them and they differ tremendously in the number of elements in their respective groups.<p>As far as I know the Langlands programme is done: all possible forms have been found. Thus since Mother Nature likes, for example Linear algebra, since it follows it in QM, and since sporadic groups are applicable to linear algebra maybe there&#x27;s new physics there. The Standard Model is built-on Linear Algebra well in the case of non-countable infinite groups ... so the tantalizing question is: are there hidden gems in algebra of sporadic groups which we are less familiar?