School of Mathematics

Johannes Siemons

Recent Research: Preprints & Publications

This page is written to give you an idea in what kind of mathematics I am interested in. If you are planning to become a research student you are particularly welcome; please read the next section or email me.

On this page you find

(1) Current Research,

(2) Projects for Research Students,

(3) Publications and Preprints,

(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.

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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.

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Publications and Preprints

P: Permutation Groups and Representation Theory

[P1] Connecting the permutation representations of a group,
Journal of Combinatorial Mathematics and Computing, 22 (1996) 23-31.
X
Abstract

[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

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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

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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

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Abstracts

P: Permutation Groups and Representation Theory

[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.

[P2] Abstract on "On a conjecture of Foulkes". Suppose that $\Omega = \{1,2, \ldots , ab\}$ for some non-negative integers $a$ and $b$. Denote by $P(a,b)$ the set of unordered partitions of $\Omega$ into $a$ parts of cardinality $b$. In this paper we study the decomposition of the permutation module $ \cn P(a,b)$ where $\cn$ is the field of complex numbers. In particular, we show that $\cn P(3,b)$ is isomorphic to a submodule of $\cn P(b,3)$ for $b \geq 3$. This settles the next unproven case of a conjecture of Foulkes.

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

If $G$ be a group, $H$ a subgroup of $G$ and $\Omega$ a transitive $G$-set we ask under what conditions one can guarantee that $H$ has a regular orbit (=of size $|H|$) on $\Omega$. Here we prove that if $PSL(n,q)\subseteq G\subseteq PGL(n,q)$ and $H$ is cyclic then $H$ has a regular orbit in every non-trivial $G$-set (with few exceptions). This result is obtained via a mixture of group theoretical and ring theoretical methods: Let $R$ be the ring of all $n\times n$ matrices over the finite field $F$ and let $Z$ be the subring of scalar matrices. We show that if $A$ and $M$ are proper subrings of $R$ containing $Z$, and if $A$ is commutative and semisimple, then there exists an element $x\in SL(n,F)$ such that $xAx^{-1}\cap M =Z$ or $n=2=|F|$.

[P4] Abstract on "An orbit theorem". We prove a theorem about orbits and tactical decompositions in locally finite posets. The result depends on the notion of {\it surjectivity index} in a poset. There are applications to simplicial complexes, graphs and amalgamation classes, including a result on the number of isomorphism classes of induced subgraphs in a graph.

[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 $G$.

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T: Topological Methods for Combinatorics and Permutation Groups

[T1] Abstract on "On the modular theory of inclusion maps and group actions". Let $\Omega$ be a finite set of $n$ elements, $R$ 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 and show that any interval of (*) not including $M_{n/2}$ or $M_{(n+1)/2}$ respectively, is $p$-exact for any prime $p>0$. This result can be extended to various submodules and quotient modules, and we give general constructions for permutation groups on $\Omega$ of order not divisible by $p$. If an interval of (*), or an equi- valent sequence arising from a permutation group on $\Omega$, does include the middle term then proper homologies can occur. In these cases we have determined all corresponding Betti numbers. A further application are $p$-rank formulae for orbit inclusion matrices.

[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.

[T3] Abstract on "Kernels of modular inclusion maps". We investigate the {\it inclusion map} $\partial: M_k\rightarrow M_{k-1}$ where $M_k$ is the vector space with basis formed by the $k$-element subsets of a set. In non-zero characteristic this map has interesting and the purpose of this note is to study generators for the homology modules when $R$ is a field of characteristic $p\geq n$ .

[T4] Abstract on "On modular homology in the Boolean algebra II". Let $R$ be an associative ring with identity and $\Omega$ an $n$-element set. For $k\le n$ consider the $R$-module $M_k$ with $k$-element subsets of $\Omega$ as basis. The {\it r-step inclusion map} $\partial_{r} :M_k\rightarrow M_{k-r}$ is the linear map defined on this basis through $\partial_{r} (\Delta):= \Gamma _1 + \Gamma _2 +...+ \Gamma_{k\choose r}$ where the $\Gamma _i$ are the $(k-r)$-element subsets of $\Delta$. For $m<r$ one obtains chains $${\cal M}:\,\,0\stackrel{\partial_r} {\longleftarrow} M_m\stackrel{\,\partial_r} {\longleftarrow} M_{m+r}\stackrel {\,\partial_r} {\longleftarrow} M_{m+2r}\stackrel{\,\partial_r}{\longleftarrow} M_{m+3r} \stackrel{\,\partial_r} {\longleftarrow}...\stackrel{\,\partial_r}{\longleftarrow} 0 $$ of inclusion maps which have interesting homological properties if $R$ has characteristic $p>0$. In \cite{Finite, Infin} we have introduced the notion of $p${\it -homology} to examine such sequences when $r=1$ and here we continue this investigation when $r$ is a power of $p$. We show that any section of ${\cal M}$ not containing certain {\it middle terms} is $p$-exact and we determine the homology modules for such middle terms. Numerous infinite families of irreducible modules for the symmetric groups arise in this fashion. Among these modules examples of the {\it semi-simple inductive systems} discussed in \cite{Comspl} appear and in the special case $p=5$ we obtain the {\it Fibonacci representations} of \cite{Fib}. There are also applications to permutation groups of order co-prime to $p$, resulting for example in {\it Euler-Poincar\'{e}} equations for the number of orbits on subsets of such groups.

[T5] Abstract on "On modular homology in the Boolean algebra III". Let $F$ be a field of characteristic $p$ and if $\Omega$ is an $n$-set let $M^{n}$ be the vector space over $F$ with basis $2^{\Omega}$. We continue the investigation of modular homological $S_{n}$-representations which arise from the {\it $r$-step inclusion map}. This is the $FS_{n}$-homomorphism $\partial_r : M^n \rightarrow M^n$ which sends a $k$-element subset $\Delta \subseteq \Omega$ onto the sum of all $(k-r)$-element subsets of $\Delta$. Using homological methods one can give explicit character and dimension formulae.

[T6] Abstract on "On modular homology in projective space:" For a vector space $V$ over $GF(q)$ let $L_k$ be the collection of subspaces of dimension $k$. When $R$ is a field let $M_k$ be the vector space over it with basis $L_k$. The {\it inclusion map} $\partial:M_k\rightarrow M_{k-1}$ then is the linear map defined on this basis via $\partial (X):=\sum Y$ where the sum runs over all subspaces of co-dimension $1$ in $X$. This gives rise to a sequence $${\cal M}:\,\,0\leftarrow M_0\leftarrow M_{1}\leftarrow ...\leftarrow M_{k-1}\leftarrow M_{k} \leftarrow... $$ which has interesting homological properties if $R$ has characteristic $p>0$ not dividing $q$. Following on from earlier papers we introduce the notion of $\pi${\it -homological}, $\pi${\it -exact} and {\it almost} $\pi${\it -exact} sequences where $\pi=\pi(p,q)$ is some elementary function of the two characteristics. We show that ${\cal M}$ and many other sequences derived from it are almost $\pi$-exact. From this one also obtains an explicit formula for the Brauer character on the homology modules derived from ${\cal M}$. For infinite dimensional spaces we give a general construction which yields $\pi$-exact sequences for finitary ideals in the group ring $RP\Gamma L(V)$.

[T7] Abstract on "On modular homology of 1-shellable complexes".We completely describe the $p$-modular homology of $1$-shellable simplicial complexes.

[T8] Abstract on "On modular homology of simplicial complexes: Shellability". For a simplicial complex $\Delta$ and coefficient domain $F$ let$F\Delta$ be the $F$-module with basis $\Delta$. We investigate the {\it inclusion map} given by $$\partial: \,\,\,\tau\mapsto\sigma_{1} +\sigma_{2} +\sigma_{3}+\ldots+\sigma_{k} $$ which assigns to every face $\tau$ the sum of the co-dimension $1$ faces contained in $\tau$. When the coefficient domain is a field of characteristic $p>0$ this map gives rise to homological sequences. We investigate the homology of such sequences for shellable complexes and prove that it often vanishes in all but one position. A generalization to shellable complexes of a well-known $p$-rank formula of Frankl and Wilson is a corollary.

[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.

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R: Reconstruction Indices for Finite and Infinite Permutation Groups

[R1] Abstract on "On the reconstruction of linear codes". For a linear code over $GF(q)$ we consider two kinds of `subcodes' called {\it residuals} and {\it punctures}. When does the collection of residuals or punctures determine the isomorphism class of the code? We call such a code {\it residually} or {\it puncture reconstructible}. We investigate these notions of reconstruction and show that, for instance, selfdual binary codes are puncture and residually reconstructible. A result akin to the edge reconstruction of graphs with sufficiently many edges shows that a code whose dimension is small in relation to its length is puncture reconstructible.

[R2] Abstract on "On the reconstruction index of permutation groups I: Semiregular groups". The subject of this paper is an invariant which is defined for arbitrary group actions. For this we need the following notions. Let $(G,\,\Omega)$ be an action. Then $G$ acts naturally on $\{\,\Delta\,:\,\Delta\subseteq \Omega\}$ by setting $G\ni g:\Delta\mapsto \Delta^g:= \{\,\,\delta^g\,\, :\,\,\delta\in \Delta\,\,\}$. Two sets $\Delta,\Gamma\subseteq \Omega$ are called {\it isomorphic}, denoted $\Delta\approx\Gamma$, if they are in the same $G$-orbit and they are called {\it hypomorphic}, denoted $\Delta\sim\Gamma$, if there exists a bijection $h: \Delta \rightarrow \Gamma$ so that $\,\,\forall\delta \in \Delta$ we have $\Delta \setminus \{ \delta\} \approx \Gamma \setminus \{ h(\delta) \}$. Then $\Delta$ is {\it reconstructible}if all sets hypomorphic to $\Delta$ are isomorphic to $\Delta$. The {\it reconstruction index} $\rho (G,\Omega)$ now is the least integer $r$ so that every finite$\Omega$-subset of $r$ or more elements is reconstructible. If no such $r$ exists put $\rho (G,\Omega)=\infty$. In this paper we determine the reconstruction index of all semiregular permutation groups. It is shown that $3\leq \rho (G,\Omega)\leq 5$ with a full classification in each case.

[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.
 
 

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