An Introduction to Representation Theory
Katie Steckles
Mathematician Masaki Kashiwara has been awarded the Abel Prize for 2025, “for his fundamental contributions to algebraic analysis and representation theory, in particular the development of the theory of D-modules and the discovery of crystal bases.” The prize was announced on 26 March, and the award ceremony will take place in Oslo on 20 May 2025.
It would take a while to explain all of the different aspects of Kashiwara’s work, although an overview is given on the Abel Prize website alongside the prize announcement. But one aspect that is key to the work Kashiwara has done is something I found particularly interesting when I studied mathematics at university, and that is representation theory.
I have previously written here about group theory, and representation theory is closely related. Recall that a group is a collection of objects, called elements, which can be combined together to get other elements of the group (along with some other technical rules to make everything work nicely). Simple examples include numbers (e.g. whole numbers, combined using addition), but also other more abstract ideas like permutations (ways to rearrange a set of objects, which are combined by performing one after the other).
Some groups crop up in multiple places. For example, we can consider the group of symmetries of a triangle: reflections and rotations, which can be combined by performing them consecutively, with the result being a different symmetry of the triangle. But if we keep track of these symmetries by numbering the corners of the triangle, it becomes evident that they are equivalent to the six different permutations of the numbers 1, 2 and 3.
We can also define cyclic groups – finite sets of numbers combined using modular arithmetic. For any size of group n, we can consider the set of numbers from 0 to (n-1), combined using addition, but wrapping around so if the sum of two numbers is more than n, we take the remainder modulo n. This means we can create a cyclic group of any size we want, and it is usually denoted ℤ/nℤ, or ℤn for short.
Another mathematical object we can build groups from is a matrix: a 2-dimensional array of numbers, drawn from some underlying chosen set of numbers (for example, a matrix might contain real numbers, or be restricted to only whole numbers, or just zeroes and ones, working modulo 2). We can combine matrices by matrix multiplication, and this allows us to define matrix groups.
To meet the full criteria of being a group, we need to restrict ourselves to specific sets of matrices – groups need elements to have inverses, and for it to be possible to combine them together in different orders. This means we need the matrices we are working with to be invertible, and to be square. (If you do not already know about how matrix multiplication works, or what an invertible matrix is, do not worry too much about it – you can consider the matrices just to be objects we can combine in the right kind of ways).
For example, the set of n-by-n invertible matrices with entries from the real numbers forms a group – called GL(n, ℝ), or the ‘general linear group of degree n’. This is an infinite group, as it has infinitely many elements, and we can consider subsets of this group that form subgroups – for example, the group made from matrices from this set which have determinant 1 form a subgroup, as multiplying two of them together will always give another matrix with determinant 1.
Time to Represent
Representation theory is an exceptionally powerful idea, which fundamentally involves using a group of matrices to represent another group. It turns out that whatever the group is you are considering, there is a way to represent it as a collection of matrices, which themselves form a group structure that maps exactly onto the group you are looking at.
For example, our triangle symmetries and equivalent permutations from earlier can be represented using a group of permutation matrices: these are simply variations on the identity matrix with the columns shuffled around as per the given permutation. (The matrices shown here match the permutations above, from left to right).
For another example, consider the following four matrices:
If you pick a pair of these and multiply them together, you will just get another matrix from this set. It follows all the rules of a group – each of the matrices is invertible, and its inverse is also in the group. The group that this matrix group mirrors the structure of is called the Klein Four Group. This consists of four elements, and has the structure ℤ₂ × ℤ₂ – that is, it is made from two copies of the group ℤ₂, which itself consists of the numbers 0 and 1 only. This means it is different from the cyclic group ℤ4 – this also contains four elements, but they behave differently.
The Klein Four group can be thought of as being made from pairs of elements from the two copies of ℤ₂, which can be denoted (0,0), (0,1), (1,0) and (1,1), and are combined by adding the entries pairwise and reducing modulo 2 (so, (1,0) + (1,1) would give (0,1).) The four matrices above each correspond to one of these pairs, and behave in exactly the same way when combined.
Matrix groups do not even have to be that complicated – a 1-by-1 matrix is still a square matrix, and if we want to represent a cyclic group of n elements, we could do it boringly by saying that it is the group of 1-by-1 matrices, using the underlying number set as ℤn+1 (and then not including 0). Multiplying together 1-by-1 matrices is just the same as multiplying together the numbers (and taking the result modulo n). Not only is this a boring approach, it also does not work unless n is a prime number, since we need the underlying number set to have the right properties to make the matrix multiplication work properly.
Another option is to let the underlying number set be ℂ, the complex numbers – and find the numbers which are the nth roots of unity. For example, the complex number i is a fourth root of unity, and multiplying it by itself repeatedly gives -1, -i and 1. The subset of GL(1,ℂ) consisting of 1-by-1 matrices containing these four values is a way to represent the cyclic group of order four, ℤ4, and we can use a similar trick to represent any cyclic group.
Representation theory is incredibly useful, as once something is being represented by matrices, it allows us to apply well-known and widely-used techniques from linear algebra. As I described in a previous blog post, this is one of the incredibly powerful aspects of mathematics – we do not have to re-invent techniques we already have, as representation theory gives us a bridge between different areas of mathematics.
Kashiwara’s work was in algebraic analysis, which combines analysis – a field of mathematical techniques conventionally used to study functions, and how things change over time – with algebra, including ideas from representation theory.
Partial differential equations, which describe systems where variables depend on each other in different ways, are used across physics and engineering to model real-world behaviour – and are conventionally studied using analysis. But algebraic analysis, which Kashiwara worked on throughout his career, uses techniques from algebra to study these kinds of mathematical objects.
D-modules, introduced in Kashiwara’s master’s thesis, are algebraic structures related to the operation of differentiation (strictly, they are rings of differential operators), and provide a powerful link between the two areas. Kashiwara’s work has allowed mathematicians to use techniques from algebra to study differential equations, and opened up the field of algebraic analysis. The Abel prize recognises this important work, and how it has given mathematicians even more tools to understand our world.
The post An Introduction to Representation Theory originally appeared on the HLFF SciLogs blog.