Quantum Topology Seminar
Scott Carter
University of South Alabama
Quipu Decorated permutation representations of finite groups
Abstract: This is a summary of a few chapters in an upcoming book which is joint work with Yongju Bae and Byeorhi Kim. In the talk we'll concentrate upon the three non-trivial modulo 2 extensions of the group of permutations on four elements, $\Sigma_4$. They all map surjectively to six element permutation group, $\Sigma_3$. The kernels are either the quaternions or the cartesian product of three copies of the integers modulo 2. The group $\Sigma_4$ also maps surjectively to $\SIgma_3$ with its kernel being the Klein $4$-group. The other groups are the group of $(2 \times 2)$ non-singular matrices over the integers modulo $3$, the determinant $1$ $(2\times 2)$ matrices over the integers modulo $4$, and the binary octahedral group. The corresponding short exact sequences can be written as strings with quipu upon them. The quipu are found in various familiar (and not so familar) small finite groups.
We use the quipu representations to compute some $2$-dimensional cocycles.
These techniques are, perhaps, known to some, but the topological perspectives are new.
Thursday June 1, 2023 at 12:00 PM in Zoom