BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Dimitri Leemans (Université Libre de Bruxelles) DTSTART:20230202T233000Z DTEND:20230203T003000Z DTSTAMP:20260513T193636Z UID:SFUOR/12 DESCRIPTION:Title: T he Number of String C-groups of High Rank\nby Dimitri Leemans (Univers ité Libre de Bruxelles) as part of PIMS-CORDS SFU Operations Research Sem inar\n\nLecture held in ASB 10908.\n\nAbstract\nAbstract polytopes are a c ombinatorial 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 ha ve the highest level of symmetry (namely the automorphism group has one or bit on the set of maximal chains)\, they are called regular. One can then use string C-groups to study them.\n\nIndeed\, string C-groups are in one- to-one correspondence with abstract regular polytopes. They are also smoot h quotients of Coxeter groups.\n\nThey consist in a pair $(G\,S)$ where $G $ is a group and $S$ is a set of generating involutions satisfying a strin g 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 associa ted polytope.\n\n \n\nIn 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 ne cessarily the symmetric group $S_n$.\n\nWe have just proven in the last mo nths that if $n$ is large enough\, up to isomorphism and duality\, the num ber of string C-groups of rank $r$ for $S_n$ (with $r \\geq (n+3)/2$) is t he same as the number of string C-groups of rank $r+1$ for $S_{n+1}$. \n\ nThis 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-g roups of rank $(n+3)/2+k$ for $S_{n+k}$ for any positive integer $k$. \n\ nThe 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 $S_{2r -3}$.\n\nA consequence of this result is the complete classification of al l string C-groups of $S_n$ with rank $n-\\kappa$ for $\\kappa \\in {1\,\\l dots\,6}$\, when $n \\geq 2 \\kappa+3$\, which extends previous known res ults.\n\nThe number of string C-groups of rank $n-\\kappa$\, with $n \\geq 2 \\kappa +3$\, of this classification gives the following sequence of in tegers indexed by $\\kappa$ and starting at $\\kappa = 1$.\n$$\\Sigma{\\ka ppa}=(1\,1\,7\,9\,35\,48).$$\nThis sequence of integers is new according t o the On-Line Encyclopedia of Integer Sequences.\n\nJoint work with Peter J. Cameron (University of St Andrews) and Maria Elisa Fernandes (Universit y of Aveiro)\n LOCATION:https://researchseminars.org/talk/SFUOR/12/ END:VEVENT END:VCALENDAR