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
(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,
[P2] On a conjecture of Foulkes, with S Dent,
[P3] Intersections of matrix algebras and permutation representations
of PSL(n,q),
[P4] An orbit theorem, with VB Mnukhin, submitted Graphs and Combinatorics,
[P5] Incidence structures with tight automorphism groups, in preparation.
[P6] Regular orbits of cyclic subgroups in permutation representations
of
T: Topological Methods for Combinatorics
and Permutation Groups
[T1] On the modular theory of inclusion maps and group actions,
[T2] On modular homology in the Boolean Algebra (with VB Mnukhin),
[T3] Kernels of modular inclusion maps,
[T4] On modular homology in the Boolean algebra II, with S Bell and
P Jones,
[T5] On modular homology in the Boolean algebra III, with P Jones,
submitted,
[T6] On modular homology in projective space, with VB Mnukhin,
[T7] On modular homology of 1-shellable complexes,
[T8] On modular homology of simplicial complexes: Shellability,
[T9] On modular homology of simplicial complexes: Saturation,
R: Reconstruction Indices for Finite and Infinite
Permutation Groups
[R1] On the reconstruction of linear codes (with P Maynard),
[R2] On the reconstruction index of permutation groups I: Semiregular
groups
[R3] On the reconstruction index of permutation groups II: wreath
products.
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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 $G$.
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.
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.
Journal of Combinatorial Mathematics and Computing, 22 (1996)
23-31.
X Abstract
Journal of Algebra, 226 (2000) 4236-249.
X Abstract
with A Zalesskii, Journal of Algebra , 226 (2000) 451-478.
X Abstract
November 1999.
X Abstract
Y dvi
certain simple groups, with A Zalesskii, in preparation.
X Abstract
Y dvi
with VB Mnukhin, Journal of Combinatorial Theory 74 (1996)
287-300.
X Abstract
Journal of Algebra., 179 (1996) 191-199.
X Abstract
Discrete Mathematics. 174 (1997) 309-315.
X Abstract
Journal of Algebra, 199 (1998) 556-580.
X Abstract
Journal of Algebra.
X Abstract
Y dvi
Journal of Pure and Applied Algebra, 151 (2000) 51-65.
X Abstract
with VB Mnukhin, submitted.
X Abstract
Y dvi
with VB Mnukhin, in press, Journal Combinatorial Theory A, appears
2000-1.
X Abstract
Y dvi
with VB Mnukhin, submitted, Journal Combinatorial Theory A.
X Abstract
Y dvi
Journal of Combinatorial Designs, 6 (1998) 285-291.
X Abstract
(with P Maynard), submitted Journal Combinatorial Theory B, February
2000.
X Abstract
Y dvi
(with P Maynard), in preparation.
X Abstract
Y dvi