Related papers: Rank reduction of string C-group representations

We give a rank augmentation technique for rank 3 string C-group representations of the symmetric group $S_n$ and list the hypotheses under which it yields a valid string C-group representation of rank 4 thereof.

In this paper, string C-groups of all ranks $3 \leq r \leq \frac{n}{2}$ are provided for each alternating group $A_n$, $n \geq 12$. As the string C-group representations of $A_n$ have also been classified for $n \leq 11$, and it is known…

We determine the ranks of string C-group representations of the groups ${\rm PSp}(4,\mathbb{F}_q)\cong\Omega(5,\mathbb{F}_q)$, and comment on those of higher-dimensional symplectic and orthogonal groups.

We classify C-groups of ranks $n-1$ and $n-2$ for the symmetric group $S_n$. We also show that all these C-groups correspond to hypertopes, that is, thin, residually connected flag-transitive geometries. Therefore we generalise some similar…

String C-groups are precisely the automorphism groups of abstract regular polytopes. A certain regular d-polytope C_d with an automorphism group of order 2^{2d-1}, discovered by Conder and shown to have the smallest number of flags among…

We prove that for any integer $n\geq 12$, and for every $r$ in the interval $[3, \ldots, \lfloor (n-1)/2\rfloor]$, the group $A_n$ has a string C-group representation of rank $r$ therefore showing that the only alternating group whose set…

If $\Gamma$ is a string C-group which is isomorphic to a transitive subgroup of the symmetric group Sym(n) (other than Sym(n) and the alternating group Alt(n)), then the rank of $\Gamma$ is at most $n/2+1$, with finitely many exceptions…

The study of string C-group representations of rank at least $n/2$ for the symmetric group $S_n$ has gained a lot of attention in the last fifteen years. In a recent paper, Cameron et al. gave a list of permutation representation graphs of…

If $G$ is a transitive group of degree $n$ having a string C-group of rank $r\geq (n+3)/2$, then $G$ is necessarily the symmetric group $S_n$. We prove that if $n$ is large enough, up to isomorphism and duality, the number of string…

In this paper we give a non-computer-assisted proof of the following result: if $G$ is an even transitive group of degree $11$ and has a string C-group representation with rank $r\in\{4,5\}$ then $G\cong\PSL_2(11)$. Moreover this string…

This work provides a classification of string C-groups of order 2m and exponent at least 2^(m - 3). Prior to the classification, we complete the list of groups of exponent 2^(m - 3) and rank at least 3. The main result states that, aside…

We study irreducible restrictions from modules over alternating groups to subgroups. We get reduction results which substantially restrict the classes of subgroups and modules for which this is possible. This is known when the…

When the standard representation of a crystallographic Coxeter group $\Gamma$ is reduced modulo an odd prime $p$, a finite representation in some orthogonal space over $\mathbb{Z}_p$ is obtained. If $\Gamma$ has a string diagram, the latter…

This work applies the modular reduction technique to the Coxeter group of rank 4 having a star diagram with labels 5, 3, and $k = 3, 4, 5, \text{ or } 6$. As moduli, we use the primes in the quadratic integer ring $\mathbb{Z}[\tau]$, where…

Building on recent work of Robertson and Steger, we associate a C*-algebra to a combinatorial object which may be thought of as a higher rank graph. This C*-algebra is shown to be isomorphic to that of the associated path groupoid.…

We classify globally irreducible representations of alternating groups and double covers of symmetric and alternating groups. In order to achieve this classification we also completely characterise irreducible representations of such groups…

We introduce parabolic degenerations of rational Cherednik algebras of complex reflection groups, and use them to give necessary conditions for finite-dimensionality of an irreducible lowest weight module for the rational Cherednik algebra…

Motivated by the Langlands program in representation theory, number theory and geometry, the theory of representations of a reductive $p$-adic group over a coefficient ring different from the field of complex numbers has been widely…

New continuous group transforms, together with their discretization over a lattice of any density and admissible symmetry, are defined for a general compact simple Lie groups of rank $2\leq n<\infty$. Rank 1 transforms are known. Rank 2…

This survey paper aims at giving the state of the art in the study of string C-group representations of almost simple groups. It also suggest a series of problems and conjectures to the interested reader.