What is Group Theory and How Does it Relate to Representation Theory?

[matt grime]

I think I will write out the beginnings of a "book" that I'd been meaning to get around to starting at some point. Better that someone might actually read it here than trust to them finding it on my web page.

Any requests, let me know by PM (or if this is inappropriate for this forum). Chapter one will appear some time soon. In the meantime here are some group theory notes to be getting on with:

http://www.maths.bris.ac.uk/~maxmg/docs/groups.pdf

 

[matt grime]

Chapter -1 Introduction and History

(Don't be put off by any of the terms here; they will all be explained properly later)

Groups are often introduced to unwary undergraduates in ways that leave them puzzled as to what these strange things might be. This is the so-called abstract group theory, and is in many ways completely misleading for groups are very concrete things, at least as concrete as anything can be in mathematics. Let us take a simple example, and one that is in some sense the origin of group theory.
Example: Let C be the complex numbers and consider the two functions f and g that map C into itself given by
f(z)=z for all z (the identiy map)
g(z)=z* (the map sending z to z conjugate)
I think most undergraduates, and indeed well educated high school students would be able to understand those two things, and amazingly they're using groups as nature, or Galois, intended.
Admittedly in Galois's original work where groups were first mooted as objects of study he was looking at things a little bit more complicated than numbers of the form a+ib, but not a lot more complicated, and today we study this field in courses called "Galois Theory", and a very beautiful part of mathematics it is too.
In these tutorials we will look at another equally beautiful aspect of pure mathematics whose origins lie with Schur and Frobenius from circa 1900 but which are still of active research interest today: representation theory of finite groups.
There are many books about this subject (some good, many not), and each has there own way of developing the material from the classical "characteristic zero" to some other aspect of the subject. Representations use vector spaces, and when we refer to some field we are referring to the field of the vector space. Heartily recommended are
1. James and Liebeck's Representations and Characters of Groups
A very good introduction to representation theory over the reals and complex numbers with applications to chemistry in an appendix. Gives every character table you can reasonably expect to need as an undergraduate
2. Fulton and Harris's Representation theory. Goes from the complex numbers to Lie algebras
3. Isaac's Character Theory of Finite Groups. The title says it all.
We aim to have the breadth of (2) with the clarity of (1) filling in lots of material that makes (3) interesting. It's a lofty aim but a good one to fail to achieve.

The initial area we will look at is the classical (ie 100 year old) representation theory over the complex numbers that is very well understood. Where we hope to diverge from the books that this author has read is that we then intend to metamorphose into a similarly explained work when we have "more interesting" fields. As we shall see what is actually the key thing is what happens if we allow the characteristic of the field to divide the order of the group. Or more accurately we examine what happens when we fail to be able to divide by |G|. This may not seem like much but in making this extra assumption then we are led onto some very complicated maths that is still a very active research topic.

Essentially we understand things over the complex numbers, and we can use them. And I don't mean just mathematicians: chemists, physicists, engineers all use representation theory of some kind or another.

If we have something where |G| cannot be inverted then we get some rich and complicated structure that we really don't understand at all as well, but which leads to some very interesting mathematics.

The reader should be aware that the term "representation theory" is a blanket term for "representations" (ie actions of some algebraic gadget, not just a group, on a vector space) of many things. Physicists also are interested in representations of Lie Algebras and we may touch on those things too.

 

[matt grime]

Chapter 0 Preliminaries and Notational Conventions
I will be using the latex features to typeset lots of mathematics, however I will not overly abuse that since the images take time to reload into a page. Thus I will adopt the "psuedo-tex" standards for things. In particular x_r means subscript, ie [itex]x_r[/itex] and x^r means superscript [itex]x^r[/tex], hopefully I will choose whichever format is best and not cause any confusion.
0.1 Groups
Recall the bad definition of a group, G, is a set of elements with a rule for composing any two elements to obtain a third element of G that satisfies some rules. Note we already have "closure" since I have declared that composing two elements of G gives an element of G. We use juxtaposition to indicate composition, ie xy means x composed with y. The other rules are
associatvity: (xy)z=x(yz) for all x,y,z in G
identity: there is an element e satisfying ex=xe=x for all x in G
inverses: given any x in G there is some y in G satisfying xy=yx=e
Examples:
the integers, rationals, reals or complex numbers under addition.
the non-zero rationals, reals or complex numbers under multiplication
the mxn matrices over any of the above with addition
the nxn invertible matrices over any of the above under matrix multiplication.
Non-examples: ie these are NOT groups
the rationals under multiplication since 0 has no multiplicative inverse.
the invertible nxn matrices under addition (exercise find two invertible matrices that add together to give a non-invertible matrix.
What I've just done is give an abstract group definition, and this is probably the one you first saw when you did a bit of algebra for the first time. Personally I prefer to give the concrete definition but I'm going to assume that you already know what a group is. If not you could try reading my notes linked to in the first post on concrete groups.
Accepting that you already know what a group is...
Definition: a group is abelian (aka commutative) if xy=yx for any pairs of elements x and y in G. Two elements commute if xy=yx
Exercise: which of the examples are abelian and which are not?
We actually need very little group theory for the most of what we do, and any we need later will be introduced at the relevant time. The only things we need to make sure we agree on are some symbols.
List of symbols
G is a group, |G| will be its order. We are only interested in the cases when G is finite ie |G| is a natural number.
C_G(x) means the conjugacy class in G of an element x. It is the set of all elements of the form yxy^{-1}
Z(G) is the centre of G; the set of all elements that commute with every element of G.
Exercise show x is in Z(G) iff C_G(x)={x}
if x is in G we will denote by ord(x) its order; that is the smallest strictly positive integer n such that x^n=e
Recall (though we shan't prove it) that the order of any element divides the order of the group.
Finally, the only other thing we need to know right now about groups is what a homomorphism is.
Definition. If G and H are two groups are homomorphism from G to H is a map p:G-->H that sends each element in g to some element in p(g) in a way that respects the group structures:
p(xy)=p(x)p(y) and p(x^{-1})=p(x)^{-1}
these say that if we consider the operations of composition or inverting, well, I can do them in G and send the result to H or send the elements to H then do the operations but that both of these yield the same result.
0.2 Vector Spaces.
Recall that a field, F, is a set with two operations + and * such that under + F is an abelian group with identity 0 and that F\{0} is an abelian group under * with identity 1. We often refer to F\{0} (ie F without the element 0) as the multiplicative group and write it is [tex]F^{\times}[/tex]

Examples: The rational, real or complex numbers.

Further examples: Let p be a prime number and consider the set of residues mod p, ie let

F={0,1,...,p-1}

then F is a field.

Proof: that it is a group under addition is obvious, and most of the fact that its non-zero elements are a group under multiplication is too, the only non-obvious thing is multiplicative inverses. Recally Euclid's algorithm and highest common factors. If x is any element of 1,2,..,p-1 then it is coprime (considering it as an integer) to p, so there are integers y and z such that xy+pz=1 (again in the integers). So just take the residue class of y for the inverse of x.

F here is a "field of characteristic p", and since it has p elements in it, it is labelled F_p

Definition: If F is any field it contains the element 1, and thus it contains 1+1, 1+1+1, etc. If no amount of adding up of 1's ever gives 0 then we say the field has characteristic zero. If we ever manage to get 0 adding up 1's let n be the first time we get 0. We say F has characteristic n.

Lemma: if F has characteristic n>0 then n is a prime.
Proof: if n is not prime let n=ab where a and b is a non-trivial factorization (ie a and b are both different from 1 and n). Consider the elements gotten by adding up a 1's and b 1's in F. These multiply together to give 0, but we know F\{0} is a group and this is a contrdiction (not closed), so n can have no non-trivial factorizations, hence it is a prime.

Lemma: If p is a prime then there is a field of characteristic p
Proof: take the resdiue classes mod p.

Lemma: there is more than one field of any given prime characteristic.
Proof: we omit the proof, preferring instead to give an example that mimics the construction of C from R. For ease we consider the case F_2; the general case is no harder.

F_2 as a set is just {0,1}. Consider the polynomial f(x)=x^2+x+1. This has no root in F_2, ie f(0)=1 and f(1)=1, just like x^2+1 has no root in R. So, just like the R to C case we add a symbol, usually an alpha, but we'll use a instead, and consider

F=F_2[a]

F_2[a] is the set of all elements of the form x+ay just like C=R is the set of all elements x+iy.

F_2[a] has 4 elements.

0,1,a,a+1

these are indeed a group under addition as we can see by inspection recalling that 2=0 mod 2. What about multiplication? well we need to work out a^2 (a times a), but we know that a satisfies a^2+a+1=0, or a^2=a+1 (recall also that 1=-1 mod 2). Similarlry a(a+1)=a^2+a=1, and

(a+1)(a+1)
=a^2+2a+1 multiplying out
=a^2+1 since 2=0
=a from the identity a^2+a+1=0

so it is indeed a field with 4 elements.

The general idea over F_p is to find a polynomial with no roots, which is easy f(x)=x^p-x-1 will do the trick. Why? Because by Fermat's little theorem x^p=x mod p for all x, so that x^p-x=0 for all x, thus f(s)=1 for any s in {0,1,..,p-1}. Now we add some element a that formally satisfes f(a)=0 and form F_p[a] as before.

We'll end this for now, since it is getting too far from representation theory. Though I will perhaps have to do a little more about fields in char p when we come to look at them in more detail to prove to you that I'm not making any restrictions that are too strong ie imposing so many rules that nothing can possibly satisfy them all.

 

[matt grime]

I think from now on I will do these in latex and post links to pdfs or it will get completely ridiculous