The Number of String C-groups of High Rank

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 $rgeq (n+3)/2$, 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.

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.

Joint work with Peter J. Cameron (University of St Andrews) and Maria Elisa Fernandes (University of Aveiro)

Mathematics

Audience: researchers in the topic

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PIMS-CORDS SFU Operations Research Seminar

Series comments: In the "Big Data Hub", room ASB 10908.

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