Polytopes of high rank for the symmetric groups

Fernandes, Maria Elisa; Leemans, Dimitri

doi:10.1016/j.aim.2011.08.006

Cited by 41 publications

(68 citation statements)

References 13 publications

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“…Observe that similar classification results have been obtained for the Suzuki simple groups Sz(q) [11], the groups PSL(3, q) and PGL (3, q) [3], the symmetric groups [8] and the alternating groups [10,9].…”

Section: Introductionsupporting

confidence: 73%

Almost simple groups with socle PSL(2,q) acting on abstract regular polytopes

2015

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regular polytopes Thin regular geometries Almost simple groups of PSL(2, q) type Our paper deals with the classification of abstract regular polytopes for almost simple groups with socle PSL(2, q). We consider all almost simple groups PSL(2, q) ≤ G ≤ PΓL(2, q) and determine the maximal rank of string C-group representations for G, i.e. the maximal rank of an abstract regular polytope with automorphism group G, as well as the existence of string C-group representations of lower ranks. Similar results have already been obtained by various authors in the cases G ∼ = PSL(2, q) and G ∼ = PGL(2, q) and they are summarized in this paper.

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Section: Introductionsupporting

confidence: 73%

Almost simple groups with socle PSL(2,q) acting on abstract regular polytopes

2015

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“…Hence, applying Corollary 1.3 gives a family of string C-group representations for Sym(m) of every rank from m − 1 down to 3. This was already proved in [3,Theorem 3].…”

Section: Symmetric and Alternating Groupsmentioning

confidence: 53%

Rank reduction of string C-group representations

Brooksbank

Leemans

2019

Proc. Amer. Math. Soc.

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We show that a rank reduction technique for string C-group representations first used in [3] for the symmetric groups generalizes to arbitrary settings. The technique permits us, among other things, to prove that orthogonal groups defined on d-dimensional modules over fields of even order greater than 2 possess string C-group representations of all ranks 3 n d. The broad applicability of the rank reduction technique provides fresh impetus to construct, for suitable families of groups, string C-groups of highest possible rank. It also suggests that the alternating group Alt(11)-the only known group having 'rank gaps'-is perhaps more unusual than previously thought.

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“…As we pointed out before, Conder and Sjerve and Cherkassoff studied these groups in relation with regular maps. More recently Fernandes and Leemans studied the symmetric groups (see [10]), and then, together with Mixer took care of the alternating groups (see [11] and [12]). …”

Section: Regular Polytopes and Almost Simple Groupsmentioning

confidence: 99%

Chiral Polytopes and Suzuki Simple Groups

Hubard

Leemans

2014

Rigidity and Symmetry

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For each q ¤ 2 an odd power of 2, we show that the Suzuki simple group S D S z.q/ is the automorphism group of considerably more chiral polyhedra than regular polyhedra. Furthermore, we show that S cannot be the automorphism group of an abstract chiral polytope of rank greater than 4. For each almost simple group G such that S < G Ä Aut.S/, we prove that G is not the automorphism group of an abstract chiral polytope of rank greater than 3, and produce examples of chiral 3-polytopes for each such group G.

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Polytopes of high rank for the symmetric groups

Cited by 41 publications

References 13 publications

Almost simple groups with socle PSL(2,q) acting on abstract regular polytopes

Almost simple groups with socle PSL(2,q) acting on abstract regular polytopes

Rank reduction of string C-group representations

Chiral Polytopes and Suzuki Simple Groups

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