Introduction to character theory

Character theory is a popular area of math used in studying groups.

Representations. Character theory has its basis in representation theory. The idea of representation theory is that every finite group (and a lot of infinite ones!) can be represented as a collection of matrices. For instance, the group of order 2 can be represented by two nxn matrices, one (call it A) with all -1’s on the diagonal, and one (call it B) with all 1’s. Then A2 =B, B2 =B, and AB=BA=A, which shows that it really is the group of order 2.

Another way of representing the same group is to have A be the 2×2 matrix that is 0 on the diagonal and 1 elsewhere. B is still the identity. This is another perfectly good representation.

Sometimes it is helpful to look at matrices which only represent a part of a group; in this situation, you don’t have an isomorphism between the group and the matrices, but you do have a homomorphism. One example is sending the group of order 4 {1,t,t2 ,t3 } to the same matrices above, sending 1 and t2 to B and t and t3 to A.

It’s easy to see that there are infinitely many representations for every group. In fact, you can take any group of matrices A_1,…,A_n and conjugate them all by another matrix C to get CA_1C-1, …, CA_nC-1. This gives you another representation (this is also called a similarity transformation). The simplest representation for every group (called the trivial representation) sends every element to the identity matrix.

Characters. What mathematicians did is say, “Is there any better way to classify these representations?” One thing they tried was to find invariants, i.e. things that don’t change under transformations. One idea was to use the trace. The trace of a matrix is invariant under similarity transformations, i.e. conjugacies.

And so mathematicians would take a representation and find its trace. The collection of all these traces is called a character. So the character of our first representation for the group of order 2 is the map sending A to -n and B to n; while the character of the second representation is the map sending A to 0 and B to 2. The character of the trivial representation in dimension n is the map sending everything to n.

This is a very simple example, but as mathematicians tried more complicated examples, they noticed a pattern (I’m simplifying the history here). The set of characters was generated by a very small number of characters, meaning that every character was a linear combination (with positive integer coefficients) of a very small number of characters, called irreducible characters.

For instance, in the group of order 2, every character is a sum of the character with values 1,1 and the character with values -1,1. These can both be given by a representation using 1×1 matrices.

Even more interesting, this decomposition into irreducible characters always gave a decomposition of representations, meaning that the matrices could be put by a transformation into block form with each block corresponding to one of the characters.

Thus, in our first example, the nxn matrices are the sum of n copies of the 1-dimensional representation. Note that the diagonal matrices are already in block form.

For the second example, note that the character is one copy of each irreducible character. The matrix A can be conjugated to the 2×2 matrix with a 1 on the upper left and a -1 on the lower right, corresponding to the two kinds of characters.

It gets crazier. It turns out that each irreducible character is orthogonal to every other character if you write out the list and take the dot product. So, 1,1 and 1,-1 are orthogonal. Conversely, given any 2 elements of the group that are not conjugate, the corresponding lists of their values in the irreducible characters are orthogonal.

This allows one to split a character into its irreducible parts very easily.

Three random notes at the end: 1. Character values don’t have to be rational or even real, but they are always algebraic integers. 2. Abelian groups always have 1-dimensional irreducible characters. These characters are actually homomorphisms into the complex numbers. 3. Building off the previous example, you can define characters on the real line to be homomorphisms into the multiplicative complex numbers. In this case, all the characters have the form t->ei(xt), where x is a constant. These characters are still orthogonal (using integration instead of addition), and any function from the real numbers to the complex numbers can be decomposed into these characters using integrals. This is, in fact, fourier theory. The fourier transform just takes a function and splits it into its irreducible characters.

This post was originally posted on reddit.

Image by RobHar.

 

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