Recent news and events

Dimitri Leemans

Short description of activities

The test for local s-arc-transitivity has been implemented in the kernel of Magma and will be available in the next export (after release 2.18-17). We proved with Schulte and Van Maldeghem that the Ree groups in characteristic 3 have only abstract regular polytopes of rank 3 (see preprint 7). We counted with Hubard the number of chiral polytopes of rank three for a Suzuki group and thus obtained the first comparison between number of chiral and number of regular polytopes for an infinite family of simple groups. It turns out that the Suzuki groups are more chiral than regular (see preprint 1). We obtained several results on the Symmetric and Alternating groups (see published papers 1, 4, 6 and 7). With Hartley and Hubard, we built up two atlases of chiral polytopes (see published paper 3) and we studied in more details the {5,3,5} regular polytope of the first Janko group (see published paper 5). With De Saedeleer, Mixer and Pisanski, we studiet rank two coset geometries arising from edge transitive bipartite graphs (see published paper 2). With Fan, Li and Pan, we studied locally 2-arc-transitive complete bipartite graphs (see preprint 2) With Buekenhout and De Saedeleer, we finished the classification of the rank two residually weakly primitive and locally 2-transitive coset geometries of PSL(2,q) groups and studied two-transitive pairs of subgroups of PSL(2,q) (see preprints 3 and 6). With Connor, we designed a new algorithm to find apartments in coset geometries (see preprint 4). With Mixer, we designed new algorithms to search for abstract regular polytopes of a given automorphism group (see preprint 5).

Published papers

  • with A.Kiefer, On pairs of commuting involutions in Sym(n) and Alt(n), Comm. Algebra, to appear, 9 pages;
  • with J.De Saedeleer, M.Mixer and T.Pisanski, Core-free, rank two coset geometries from edge-transitive bipartite graphs, Math. Slovaca, to appear, 14 pages;
  • with M.I.Hartley and I.Hubard, Two atlases of chiral polytopes, Ars Math. Con-temp. 5 (2012), 371-382;
  • with M.-E.Fernandes and M.Mixer, All alternating groups An with n ? 12 have polytopes of rank ?(n-1)/2?, SIAM Journal on Discrete Math. 26 (2012), nr. 2, 482-498;
  • with M.I.Hartley and I.Hubard, Regular 4-polytopes from the Livingstone graph of Janko’s first group, J. Alg. Combi. 35 (2012), 193-214.
  • with M.-E.Fernandes and M.Mixer, Polytopes of high rank for the alternating groups. J. Combin. Theory Ser. A 119 (2012), 42-56;
  • with M.-E.Fernandes, Polytopes of high rank for the symmetric groups, Adv. Math. 228 (2011), 3207-3222;

Preprints

  • with I.Hubard, Chiral polytopes and Suzuki simple groups, Fields Institute Communications Series, to appear, 19 pages.
  • with W.Fan, C.H.Li and J.Pan, Locally 2-arc-transitive complete bipartite graphs, 12 pages;
  • with F.Buekenhout and J.De Saedeleer, The rank two geometries of the PSL(2, q) groups : part II. 15 pages;
  • with T. Connor, A new algorithm to find apartments in coset geometries, 14 pages.
  • with M.Mixer, Algorithms for classifying regular polytopes with a fixed automorphism group, 12 pages;
  • with F.Buekenhout and J.De Saedeleer, Two-transitive pairs in PSL(2, q). 11 pages;
  • with E. Schulte and H. Van Maldeghem, Groups of Ree type in characteristic 3 acting on polytopes, 7 pages.

Conferences

  • Symmetries of Discrete Objects 2012, Queenstown (New Zealand), Febru- ary 13-17, 2012. Magma Minicourse on Permutation Groups, 2 lectures.
  • New Zealand Mathematical Society Colloquium, Auckland (New Zealand), December 6-8, 2011. Polytopes of high rank for the symmetric groups.
  • Workshop on Symmetry, Graphs and Polytopes, Fields Institute, Toronto (Canada), October 24-27, 2011. Plenary talk. Polytopes of high rank arising from almost simple groups.
  • Seventh Slovenian International Conference on Graph Theory, Bled (Slovenia), June 20-25, 2011. Chirally regular polytopes.
  • 5th De Bru ´n Workshop on Groups, Combinatorics and Computing, NUI Galway (Ireland), April 11-16, 2011. Searching for Geometries with Magma.

Colloquium/Seminars

  • University of Auckland, Algebra Seminar, 07-10-2011. Some constructions in incidence geometry.
  • Imperial College, London Algebra Colloquium, 24-03-2011: Regular and chirally regular polytopes.