# Super-groups at odd primes

There is an extensive class of groups with the following properties:

• They have a central element “$-1$” of order two. This means that $(-1)a=a(-1)$ for all group elements $a$, and $(-1)^2=1$, the identity element.
• They have a homomorphism “det” to the group $\{\pm1\}$ with two elements such that $\det(-1)=1$.

Groups with this structure are called super-groups (by me).

Here are some examples:

• dihedral groups $D_{2m}$, where $\det$ is $-1$ on reflections and $+1$ on rotations, and with $-1$ the rotation by 180 degrees
• the quaternion group, where $-1=i^2=j^2=k^2$ and a number of choices of $\det$; similarly, generalized quaternion groups
• orthogonal groups $O(2n)$ with $-1$ the scalar matrix with $-1$ on the diagonal and $\det$ the usual determinant
• Spin and Pin groups
• so-called extraspecial 2-groups

An interesting operation on super-groups is the graded-commutative central product: $G*H$ is the group generated by elements of $G$ and elements of $H$, modulo the following relations:

• $(-1)_G = (-1)_H =: -1$
• $gh=\pm hg$, where the sign is $-1$ (the element defined above) iff $\det(g)=\det(h)=-1$

This becomes a super-group with central element of order two $-1$ and $\det(gh) = \det(g)\det(h)$.

Try to work out what $D_4 * D_4$ is!

A surprising result is that complex representations of $G*H$ are completely determined by the complex representations of $G$ and those of $H$, in an explicit but not quite straightforward way. This allows one to compute the characters of many super-groups in a very efficient way.

Super-groups come up in my current research on homotopy representations of the exotic $2$-compact group found by Dwyer and Wilkerson, and I will be happy to tell you more about this application in person.

The aim of this project, however, is to try to generalize the definition and the results I have obtained for super-groups to odd primes in the sense that $-1$ no longer has order $2$ but order $p$ for some prime $p$. This minor modification might make the theory quite different. Various avenues of research can be pursued, depending on the student’s interests and background:

• The representation theory of super-groups at odd primes, possibly including real and quaternionic representations
• The cohomology of such groups
• Interesting examples
• Fusion for such groups (fusion is a beefed-up study of subgroups of Sylow subgroups)

I have funding to employ an international Masters student from KTH as a research assistant for 6 months between now and August 2021 at 20% FTE (i.e. you will be spending 20% of your work week on this project) . The research work you will be doing cannot be used as course work or a Master’s thesis (but, a Master’s thesis could certainly build on it).

You will be working with me, in person or remotely as the situation permits, and we will have regular meetings, much like for thesis work. Your results are yours.

What I expect: You are excited about algebra, in particular groups and representations, you like to pursue not only problems given to you but also your own explorations in mathematics, you can work independently to some extent, you have good knowledge of group theory and basic representation theory (some background in algebraic topology or cohomology of groups is a bonus), and you have 20% of your time to spare without negatively affecting your course work.

What is in it for you? Besides being paid, training in doing research (on a basic level) and communicating that research with the possibility of a culmination in a publication.