## SFU Operations Research Seminars: Dimitri Leemans

- Date: 02/02/2023
- Time: 15:30

Simon Fraser University

The number of string C-groups of high rank

In this talk, we will give the latest developments on the study of string C-groups of high rank. In particular, if $G$ is a transitive group of degree $n$ having a string C-group of rank $rgeq (n+3)/2$, work over the last twelve years permitted us to show that $G$ is necessarily the symmetric group $S_n$. We have just proven in the last months that if $n$ is large enough, up to isomorphism and duality, the number of string C-groups of rank $r$ for $S_n$ (with $r \geq (n+3)/2$) is the same as the number of string C-groups of rank $r+1$ for $S_{n+1}$. This result and the tools used in its proof, in particular the rank and degree extension, imply that if one knows the string C-groups of rank $(n+3)/2$ for $S_n$ with $n$ odd, one can construct from them all string C-groups of rank $(n+3)/2+k$ for $S_{n+k}$ for any positive integer $k$. The classification of the string C-groups of rank $r \geq (n+3)/2$ for $S_n$ is thus reduced to classifying string C-groups of rank $r$ for $S_{2r-3}$. A consequence of this result is the complete classification of all string C-groups of $S_n$ with rank $n-\kappa$ for $\kappa \in {1,\ldots,6}$, when $n \geq 2 \kappa+3$, which extends previous known results. (Full abstract found at "More Details" link below.)