My name is Anton, and I do research in mathematics. Here I explain my research interests at three levels: First the layperson who knows nothing about research mathematics, then the undergraduate who maybe knows a few fundamental ideas, and then the researcher who wants to know more of the details.
For the Layperson
Mathematicians care a lot about symmetry. Partially because it is useful, but also simply because it is beautiful. We can use the symmetry of certain situations – perhaps of a geometric figure, or of something more abstract – to reduce complicated questions to much simpler ones.
So how do we study symmetry mathematically? Mathematicians have devised a kind of mathematical object which describes a collection of symmetries, which is called a group. The study of these so-called groups is known as group theory. Don't let this unassuming name fool you – these are complex and subtle objects which form a vast field of study.
A fundamental question mathematicians would like to answer is this: what are all the possible kinds of symmetry something can have? This is a bit vague, and since we need to formulate this mathematically in order to work with it precisely, we instead try to answer this (arguably identical) question: what are all the possible groups?
Unfortunately, this question seems to be completely impossible to answer in any satisfying way – groups are just too complicated. Perhaps this is a surprising fact, since symmetry seems to be so nice.
Since classifying all groups is impossible, mathematicians in the mid 20th century instead began trying to classify the finite simple groups, which (despite their name) are anything but simple! These are the groups which are, in a precise sense, the building blocks of all groups. Although it is difficult to understand how the simple groups fit together to form other groups, being able to classify all of them still tells us a lot about groups as a whole.
Fortunately, rather than being impossible, it turned out to be only nearly impossible to classify the finite simple groups. In the early 2000s, the herculean effort of classifying the simple groups was completed, forming what is (in my opinion) the greatest mathematical achievement of humankind so far.
One of the major ideas used in this enormous project was representation theory. This is a set of tools that we can use to study finite groups by interpreting a group as a collection of symmetries of space – often in dimensions much higher than three – called a representation. For reasons that still feel mysterious and magical to me, this concept has turned out to be extremely useful in studying groups, and some of the most fundamental observations that led to the classification of finite simple groups are based on representation-theoretic results. A famous example of the power of representation theory is that mathematicians had found a way to understand the representations of a group affectionately called 'the monster' before we even knew that it existed at all!
In the same way that we try to reduce the problem of classifying all groups to classifying only the simple groups, we can reduce the problem of finding all representations of a given group to finding the irreducible representations – the building blocks of all other representations. This time, the ways irreducible representations combine are much less strange than the ways groups combine, and we can often get a complete classification of representations of a group from a list of irreducible ones. Finally, a classification that is actually reasonable! This is one of the ways in which asking questions about representations leads to convenient results.
My research is about representation theory. I study special groups that were very important in the classification of finite simple groups – the groups of Lie type – because if we can answer certain questions about them, then we can answer questions about the representation theory of all finite groups. The hope is that this helps us understand how symmetry works in a deeper way someday.
For the Undergraduate
Let's say we have a group \(G\). A representation of \(G\) is a function \(\rho\) assigning to each element of \(G\) an \(n\times n\) invertible matrix with complex number entries, such that \(\rho(gh) = \rho(g)\rho(h)\). This on its own is not such an interesting definition.
We can define a notion of a sum of representations. If we can choose a basis such that \(\rho\) can always be written as a block diagonal matrix of some smaller representations \(\rho_1\) and \(\rho_2\) we say that \(\rho\) is actually the sum \(\rho_1 \oplus \rho_2\) of these smaller representations. A representation that cannot be written as such a sum is called irreducible. It can be shown that every representation decomposes as a sum of irreducible representations, or irreps for short.
Classifying the irreducible representations of a given group is a common goal when studying representation theory, as one can use a complete list of them to find out a lot of information about your group just by looking at the numbers involved. One class of groups whose representation theory we know a lot about is the symmetric groups, and there are still a lot of interesting questions to be resolved about its representations.
Here's a really cool fact. If you have a representation \(\rho\) and you take the trace of the matrices, you get what we call a character \(\chi\) of \(G\), which is in particular just a function from \(G\) into the complex numbers. Despite ignoring a huge amount of information about the representation, the character determines a representation completely – up to some appropriate notion of representations being 'the same' of course. This means that we can study characters instead of representations a lot of the time, and this turns out to be very fruitful.
Notice that in the definition I gave, I required the matrices in a representation to be over the complex numbers. This was a white lie (forgive me!) as we can actually define representations over any field we choose. Note that when I say 'field,' I mean a commutative ring with multiplicative inverses rather than a vector field or similar object.
A lot of the fundamental results of representation theory depend on two assumptions about our field:
- The field we work over is of characteristic zero. That is, \(1 + \cdots + 1\) is never equal to \(0\) in our field. If you're surprised that this is possible at all, an example of a field that is not like this is the field of integers modulo a given prime \(p\).
- The field we work over is algebraically closed. That is, every non-constant polynomial has a root in the field. It is difficult to describe examples of such fields besides the field of complex numbers, but rest assured they do exist.
Some researchers drop the first assumptions, and do modular representation theory. One of the problems that arise in this case is that not all representations decompose as the sum of irreducible ones. Instead, representations can combine in stranger ways.
My research is about keeping the first assumption, but dropping the assumption of algebraic closure. This means that I look at fields such as the rationals or the \(p\)-adic numbers, and try to inspect the representation theory over them. Sometimes there is no difference when we look at these fields. For example, the symmetric groups have no change at all in these circumstances. The smallest example of a group where there is a change is the quaternion group \(Q_8\). It's working out which groups behave differently, and how much differently, that turns out to be the hard part.
For the Researcher
I'm interested in Turull's Conjecture, which is a McKay-style conjecture in the representation theory of finite groups. Alexandre Turull proposed this in 2008 following a wealth of work in calculating certain character-theoretic invariants in a large number of groups.
The McKay conjecture (rhymes with 'McFly') is a fundamental question in the representation theory of finite groups, positing a bijection between characters of a group of degree coprime to a prime \(p\) and similar characters of the normaliser of a Sylow \(p\)-subgroup. We usually write this as there being a bijection between the sets \(\operatorname{Irr}_{p'}(G)\) and \(\operatorname{Irr}_{p'}(N_G(P))\), for a given finite group \(G\) and Sylow \(p\)-subgroup \(P\).
Following a reduction to the finite simple groups, and the work of many researchers, the McKay conjecture has been proven – although the proof involves unpublished work at the time of writing. It also bears mentioning that no reduction theorem for Turull's conjecture has been published at time of writing, but work is being done regarding it.
Turull's conjecture expects the McKay bijection not only to exist, but also to preserve a certain invariant.
Asking for the McKay bijection to preserve some invariant is not an entirely new concept. In 2004, Gabriel Navarro suggested a similar refinement stating that the McKay bijection should preserve the field of values generated by the character, over the local field \(\mathbb Q_p\). Some call this the Galois–McKay conjecture, but more often it is given Navarro's name. A reduction theorem for this conjecture has been proven.
Turull's conjecture adds an extra requirement to what Navarro suggested the bijection should preserve, namely the Schur index over the \(p\)-adics.
So, what is the Schur index? It is a perhaps unsurprising but somewhat subtle fact that even if a character takes values in a certain field (for example the real numbers) this does not mean that there is a representation over that same field which realises that character. Put as briefly as possible, the Schur index measures the failure of a character to be realised over a field generated by its character values.
I am interested in calculating Schur indices for groups of Lie type, but also more generally trying to find ways to demonstrate equality of Schur indices of characters of different groups, in the hopes that we can try and prove Turull right.
Questions
If you have questions, comments, or corrections, please do not hesitate to get in touch.