Fakulta matematiky, fyziky
a informatiky
Univerzita Komenského v Bratislave

# Algebraic Graph Theory International Webinar (10.1.2023)

## v utorok 10.3.2023 o 19:00 hod.

09. 01. 2023 15.42 hod.

The organizers of the Algebraic Graph Theory International Webinar would like to invite you to join us and other colleagues on Januar 10, 2023, at 7pm Central European Summer Time (= 6pm UTC), for the next presentation delivered by Dimitri Leemans.

The title: The number of string C-groups of high rank
Joint work with Peter J. Cameron (University of St Andrews) and Maria Elisa Fernandes (University of Aveiro)

Abstract:
Abstract polytopes are a combinatorial generalisation of classical objects that were already studied by the Greeks. They consist in posets satisfying some extra axioms. Their rank is roughly speaking the number of layers the poset has. When they have the highest level of symmetry (namely the automorphism group has one orbit on the set of maximal chains), they are called regular. One can then use string C-groups to study them. Indeed, string C-groups are in one-to-one correspondence with abstract regular polytopes. They are also smooth quotients of Coxeter groups. They consist in a pair $(G,S)$ where $G$ is a group and $S$ is a set of generating involutions satisfying a string property and an intersection property. The cardinality of the set $S$ is the rank of the string C-group. It corresponds to the rank of the associated polytope.

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 $r\geq (n+3)/2$, 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. The number of string C-groups of rank $n-\kappa$, with $n\geq 2\kappa +3$, of this classification gives the following sequence of integers indexed by $\kappa$ and starting at $\kappa = 1$. $$\Sigma{\kappa}=(1,1,7,9,35,48).$$ This sequence of integers is new according to the On-Line Encyclopedia of Integer Sequences.

The Zoom link for this semester is:
https://cuaieed-unam.zoom.us/j/87193320713?pwd=cHpiWUtYWlUvWHZjdGZteSt1QmZ5UT09
Meeting ID: 871 9332 0713
Passcode: 653250

Further details may be found at http://euler.doa.fmph.uniba.sk/AGTIW.html

where you can also find the slides and the recordings of our previous presentations. Also, if you wish to advertise an AGT friendly conference on this page, please send us the link.

Hoping to see you at the webinar, and wishing you all the best.