What is Representation Theory and how is it used? Oxford Mathematics Research investigates

Oxford Mathematician Karin Erdmann specializes in the areas of algebra known as representation theory (especially modular representation theory) and homological algebra (especially Hochschild cohomology). Here she discusses her latest work.

"Roughly speaking, representation theory investigates how algebraic systems can act on vector spaces. When the vector spaces are finite-dimensional this allows one to explicitly express the elements of the algebraic system by matrices, hence one can exploit linear algebra to study 'abstract' algebraic systems. In this way one can study symmetry, via group actions. One can also study irreversible processes. Algebras and their representations provide a natural frame for this.

An algebra is a ring which also is a vector space such that scalars commute with everything. An important construction are path algebras: Take a directed graph $Q$, which we call a quiver, and take a coefficient field $K$. Then the path algebra $KQ$ is the vector space over $K$ with basis all paths in $Q$. This becomes an algebra, where the product of two basis elements is either its concatenation if this exists, or is zero otherwise.

Algebras generalize groups, namely if we start with a group, we get naturally an algebra: take the vector space with basis labelled by the group, and extend the group multiplication to a ring structure.

When the coefficients are contained in the complex numbers, representations of groups have been studied for a long time, and have many applications. With coefficients in the integers modulo $2$, for example, the algebras and their representations are much harder to understand. For some groups, the representations have 'finite type'. These are well-understood but almost always they have 'infinite type'. With a few exceptional 'tame' cases, these are usually 'wild', that is there is no hope of a classification of the representations.

The same cases occur precisely for modulo 2 arithmetic and when the symmetry is based on dihedral or semidihedral or quaternion 2-groups. Dihedral 2-groups are symmetries of regular $n$-gons when $n$ is a power of 2. The smallest quaternion group is the famous one discovered by Hamilton.

Viewing these symmetries from groups in the wider context of algebras was used (a while ago) to classify such tame situations. Recently it was discovered that this is part of a much larger universe. Namely one can construct algebras from surface triangulations, in which the ones from the group setting occur as special cases.

One starts with a surface triangulation, and constructs from this a quiver, that is, a directed graph: Replace each edge of the triangulation by a vertex, and for each triangle

where in the last case $a=c\neq b$. At any boundary edge, draw a loop.

For example, consider triangulation of the torus with two triangles, as shown below. Then there are, up to labelling, two possible orientations of triangles and two possible quivers:

The tetrahedral triangulation of the sphere

gives rise to several quivers, depending on the oreintation of each traingle, for example:

 

The crystal in the north wing of the Andrew Wiles Building, home of Oxford Mathematics (image drawn above) can be viewed as a triangulation of a surface with boundary. We leave drawing the quiver to the reader.

Starting with the path algebra of such a quiver, we construct algebras by imposing explicit relations, which mimic the triangulation. Although the quiver can be arbitrarily large and complicated, there is an easy description of the algebras. We call these 'weighted surface algebras.' This is joint work with A. Skowronski.

We show that these algebras place group representations in a wider context. The starting point is that (with one exception) the cohomology of a weighted surface algebra is periodic of period four, which means that these algebras generalize group algebras with quaternion symmetry.

The relations which mimic triangles can be degenerated, so that the product of two arrows around a triangle become zero in the algebra. This gives rise to many new algebras. When all such relations are degenerated, the resulting algebras are very similar to group algebras with dihedral symmetry. If we degenerate relations around some but not all triangles, we obtain algebras which share properties of group algebras with semidihedral symmetry. Work on these is in progress."