## Johannes Siemons

Recent Research: Preprints & Publications

(4) Abstracts.

Research:

My main interests lie in combinatorics and group theory. For my work on permutation groups (usually finite) I am interested in linear representations of permutation groups and generic permutational properties, see [P1, P2, T1, T4, T5]. By this I mean permutational properties of the 'natural' permutation presentation of a group which are inherited to arbitrary actions of the group: Examples involve numbers of orbits of subgroups and orbits with particular properties. The finite general linear groups have been dealt with in [P3, P6] with Alex Zalesskii and it appears that the remaining classes of doubly transitive groups may now be done by similar techniques.

A more general interest are algebraic and topological methods in combinatorics. We use ideas from algebraic topology to introduce new invariants for partially ordered sets which occur essentially only in characteristic p>0 situations. The modular homology of a poset or simplicial complex is such a new invariant. It has deep connections to ordinary and Hochschild homology which are not yet fully explored. There are also connections to quantum groups, see the recent work of Kassel and Dubois-Violette. In the two most recent papers [T8, T9] a general embedding theorem is proved for the homology of all shellable complexes. On the basis of this result the modular homology of many general combinatorial objects are now known. This includes well-behaved simplicial complexes such as Coxeter complexes and Tits buildings. Other key words are Cohen-Macaulay posets and Steinberg representations.

The reconstruction index of a permutation group is probably one of the less well-known permutational properties, yet it is of big interest in combinatorics. To work out this index for a specific finite group is computationally very involved, even if the degree or order of the group is quite small. To determine the index for whole classes of groups, such as symmetric groups acting on pairs, or for certain infinite groups such as the automorphism group of the random graph, would solve several long-standing problems in combinatorics. Among these are Ulam's graph reconstruction conjectures, hence the name reconstruction index. How far this can be taken with elementary methods is not clear and in [R2, R3] we are still at the "collecting specimens" stage. In [R2] we have determined the reconstruction index of all regular groups and in [R3] we deal with imprimitive groups.

Ideas and Projects for Research Students:

If you contemplate starting research in pure mathematics then here are some helpful ideas, I hope. In each of the three areas outlined above there are interesting research projects. Typically these do not require lengthy preparations and often a third level course in algebra, group theory, combinatorics or topology course is quite sufficient for back ground. It is possible to state rewarding problems early on and I believe that creative ideas are more important than slog. More difficult problems are often quite close by and so most projects can be adjusted to whatever level is required. I greatly enjoy working with graduate students and some of the publications listed in the next section are the product of collaborations with graduate students. Please email me if you want to discuss your ideas with me.

Publications and Preprints

P: Permutation Groups and Representation Theory

[P2] On a conjecture of Foulkes, with S Dent,
Journal of Algebra, 226 (2000) 4236-249.
X Abstract

[P3] Intersections of matrix algebras and permutation representations of PSL(n,q),
with A Zalesskii, Journal of Algebra , 226 (2000) 451-478.
X Abstract

[P4] An orbit theorem, with VB Mnukhin, submitted Graphs and Combinatorics,
November 1999.
X Abstract
Y dvi

[P5] Incidence structures with tight automorphism groups, in preparation.

[P6] Regular orbits of cyclic subgroups in permutation representations of
certain simple groups, with A Zalesskii, in preparation.
X Abstract
Y dvi

T: Topological Methods for Combinatorics and Permutation Groups

[T1] On the modular theory of inclusion maps and group actions,
with VB Mnukhin, Journal of Combinatorial Theory 74 (1996) 287-300.
X Abstract

[T2] On modular homology in the Boolean Algebra (with VB Mnukhin),
Journal of Algebra., 179 (1996) 191-199.
X Abstract

[T3] Kernels of modular inclusion maps,
Discrete Mathematics. 174 (1997) 309-315.
X Abstract

[T4] On modular homology in the Boolean algebra II, with S Bell and P Jones,
Journal of Algebra, 199 (1998) 556-580.
X Abstract

[T5] On modular homology in the Boolean algebra III, with P Jones, submitted,
Journal of Algebra.
X Abstract
Y dvi

[T6] On modular homology in projective space, with VB Mnukhin,
Journal of Pure and Applied Algebra, 151 (2000) 51-65.
X Abstract

[T7] On modular homology of 1-shellable complexes,
with VB Mnukhin, submitted.
X Abstract
Y dvi

[T8] On modular homology of simplicial complexes: Shellability,
with VB Mnukhin, in press, Journal Combinatorial Theory A, appears 2000-1.
X Abstract
Y dvi

[T9] On modular homology of simplicial complexes: Saturation,
with VB Mnukhin, submitted, Journal Combinatorial Theory A.
X Abstract
Y dvi

R: Reconstruction Indices for Finite and Infinite Permutation Groups

[R1] On the reconstruction of linear codes (with P Maynard),
Journal of Combinatorial Designs, 6 (1998) 285-291.
X Abstract

[R2] On the reconstruction index of permutation groups I: Semiregular groups
(with P Maynard), submitted Journal Combinatorial Theory B, February 2000.
X Abstract
Y dvi

[R3] On the reconstruction index of permutation groups II: wreath products.
(with P Maynard), in preparation.
X Abstract
Y dvi

Abstracts

[P1] Abstract on "Connecting the permutation representations of a group". Suppose that a finite group $G$ acts on two sets $X$, $Y$ and that $FX$, $FY$ are the natural permutation modules for a field $F$. We examine conditions which imply that $FX$ can be embedded in $FY$. For primitive groups we show that there is always such an embedding when the stabilizer of a point in $Y$ has a 'maximally overlapping' orbit on $X$. For groups of rank three and four various condition in terms of orbital matrices of $G$ are investigated.

[P3] Abstract on "Intersections of matrix algebras and permutation representations of $PSL(n,q)$".

[P6] Abstract on "Regular orbits of cyclic subgroups in permutation representations of certain simple groups". In this paper we are interested in permutation representations of finite simple groups $G$ which admit a doubly transitive \pe representation. The following is shown: Apart from a short list of exceptions every cyclic subgroup $H\subset G$ has a regular orbit in any non-trivial \pe \rep of $G$.

T: Topological Methods for Combinatorics and Permutation Groups

[T2] Abstract on "On modular homology in the Boolean Algebra". Let $\Omega$ be a set, $R$ a ring of characteristic $p>0$ and denote by $M_k$ the $R$-module with $k$-element subsets of $\Omega$ as basis. The {\it set inclusion map } $\partial: M_{k}\rightarrow M_{k-1}$ is the homomorphism which associates to a $k$-element subset $\Delta$ the sum $\partial(\Delta)=\Gamma_1+\Gamma_2+...+\Gamma_k$ of all its $(k-1)$-element subsets $\Gamma_i$. In this paper we study the chain $$(*)\,\, 0\leftarrow M_{0}\leftarrow M_{1}\leftarrow M_{2} \leftarrow ... \leftarrow M_{k} \leftarrow M_{k+1} \leftarrow M_{k+2} \leftarrow ....$$ arising from $\partial$. We introduce the notion of p-exactness for a sequence. If $\Omega$ is infinite we show that (*) is $p$-exact for all prime characteristics p>0. This result can be extended to various submodules and quotient modules, and we give general constructions arising from permutation groups with a finitary section. Two particular applications are the following: The orbit module sequence of such a permutation group on $\Omega$ is p-exact for every prime p, and we give a formula for the $p$-rank of the orbit inclusion matrix if the group has finitely many orbits on $k$-element subsets.

[T9] Abstract on "On modular homology of simplicial complexes: Saturation". Among the shellable complexes a certain class is shown to have maximal modular homology. These are the so-called {\it saturated} complexes. We prove that Coxeter complexes and buildings are saturated.

R: Reconstruction Indices for Finite and Infinite Permutation Groups

[R3] Abstract on "On the reconstruction index of permutation groups II: Wreath products". We obtain various bounds for the reconstruction index of primitive and imprimitive permutation groups. The main interest are the imprimitive action of wreath products which are shown to provide examples of transitive actions with maximal reconstruction index.