Department of Economics, Mathematics and Statistics

# Research

If you click on a link to a paper you'll be taken to a page with my publication list such that the relevant paper will be the one right at the top of the page.

### Research Interests

Product-free sets: A sum-free set of integers is a set S having the property that the sum of any two (not necessarily distinct) elements of S does not lie in S. This can be generalised to any group (at which point we refer to product-free sets). Two natural questions are: what is the maximum cardinality of a product-free set in a given group G, and what are the cardinalities of maximal by inclusion product-free sets in G (we call these locally maximal nowadays, because we realised that almost everyone in the literature on sum-free and product-free sets uses "maximal" to mean "maximal by cardinality").

In this 2009 paper with Michael Giudici, I investigated small locally maximal product-free sets. We classified all groups with maximal (by inclusion) sum free sets of size 1 and 2, and made some progress on size 3 and above. It was the last paper I submitted before going on maternity leave with my first child, so I had to leave the problem for a while! My PhD student Chimere Anabanti and I took up the work again in 2014, and completed the classification of size 3 locally maximal product-free sets here.

We have also looked at so-called "filled" groups. Street and Whitehead observed that in an elementary abelian 2-group a product-free set S is locally maximal if and only if G is the union of S and SS. That is, every element of G is either in S or is the product of two (not necessarily distinct) elements of S. We say that S "fills" G. This does not happen in general, for example in the quaternion group of order 8, the unique involution forms a locally maximal product-free set. Street and Whitehead define a group as "filled" if every locally maximal product-free set fills the group.  Therefore the elementary abelian 2-groups are a collection of examples of filled groups. Street and Whitehead classified the filled abelian groups - the only other examples are the cyclic groups of order 3 and 5, and they made a conjecture about filled and non-filled dihedral groups. Chimere and I have investigated filled non-abelian groups here. We disproved the conjecture, and we also classified all filled groups of odd order (the only ones are the cyclic ones of order 3 and 5). We further showed that any filled nilpotent group is either cyclic of order 3 or 5, or a 2-group. here we classified the filled dihedral groups. Therefore it is of interest to classify the filled 2-groups. We know that no dihedral 2-groups or generalised quaternion 2-groups are filled except for the dihedral group of order 8 (and the one of order 4, if you count it as dihedral). We suspect the answer involves extraspecial groups.

Coxeter Groups: Nearly all my work on Coxeter groups is joint with Peter Rowley, in a longstanding collaboration that goes back to the dawn of prehistory (i.e. my PhD).

• We have three papers (here, here and here) on the idea of "excess" in Coxeter groups. It is a well known result of Carter that any element w of a finite Coxeter group W can be expressed as a product of at most two involutions. But there is no guarantee that this expression will be reduced (in the sense that it is a minimal length expression for w as a product of the fundamental reflections that generate W). We define the "excess" of w, written e(w), to be the minimum value of l(x) + l(y) - l(w) over all x, y with w = xy and x² = y² = 1. So any involution has excess 0, for example, as does any element which does have a reduced expression as the product of two involutions. In our papers we show that every element of an arbitrary Coxeter group that can be written as a product of two involutions is conjugate to an element of excess zero; we also show that in a finite group there is at least one element of minimal length and excess zero, and at least one element of maximal length and excess zero, in any conjugacy class. Strangely, and surprisingly, we discovered that excess is not necessarily preserved on descent to a standard parabolic subgroup.
• We have also looked at involution classes in Coxeter groups. Here we obtain polynomials f(t) where the coefficient of t^l is the number of involutions of length l in a given conjugacy class of involutions, for a finite Coxeter group W. These generalise results for the symmetric group. Here we give a method of calculating the minimal and maximal length of involutions in a conjugacy class of a Coxeter Group, equipped with an (arbitrary) element of that class. Then here we give explicit descriptions of the sets of minimal and maximal length involutions in the conjugacy classes of finite (irreducible) Coxeter groups.
• The notion of the length of an element in a Coxeter group W is, and has been, of fundamental importance in the study of Coxeter groups. Here, here, here, here, here and here (!) such ideas have been extended to assign a length, called the Coxeter length, to all subsets of W. A number of results have been obtained which suggest that the Coxeter length of subsets will be of value in future investigations into Coxeter groups. This research was supported by EPSRC, who awarded me a Postdoctoral Research Fellowship for the work. A survey covering most of the work so far on Coxeter Length is here. On a related but different note, this paper looks what we call negative orbits - that is, for a given positive root α and subset X of W, the intersection of X.α with the set of negative roots. X is usually a conjugacy class so this isn't really an orbit strictly speaking. Some quite interesting things happen nevertheless.
• My Ph.D. thesis, (Coxeter groups, conjugacy classes and relative dominance, 2000) was largely concerned with the lengths of elements in conjugacy classes of Coxeter groups. Let W be a Coxeter group with a distinguished generating set R. The set of reflections is defined to be {wrw-1 | r in R, w in W}. Let w be an element of W. Then w is called bad upward if it does not have maximal length in its conjugacy class, but for all reflections s, l(sws) is at most l(w). A flat conjugacy class of W is a class whose elements all have the same length. An element w of W is called an evil element if it does not lie in a flat class, but, for all reflections s, l(sws) = l(w). Among the results proved were:(i) Let W be an infinite irreducible Coxeter group. Suppose w is an element of W such that for all reflections s in W, l(sws) does not exceed l(w). Then w is in a flat class of W [article]. This result implies that are no bad upward or evil elements in an infinite irreducible Coxeter group. (ii) A Coxeter group has no evil elements [article].
• I also (briefly) investigated dominance in the root systems of Coxeter groups here. (Brink and Howlett introduced the idea of dominance of roots used it to show that Coxeter groups are automatic.)

Commuting Graphs: Suppose G is a finite group and X is a subset of G. The commuting graph on the set X, which we denote by C(G,X), has X as its vertex set with x, y in X being joined by an edge whenever xy = yx. If X consists entirely of involutions, then we call C(G,X) a commuting involution graph. Many authors have studied C(G,X) for different choices of G and X, and from a number of different perspectives. For example in the seminal paper of Brauer and Fowler this graph is studied in the case when G has even order and X = G {1}. Typical of a number of results obtained is that in a group with more than one conjugacy class of involutions any two involutions are distance at most 3 apart in C(G,X). Commuting involution graphs arose in the work of Fischer during his investigation of so-called 3-transposition groups, one outcome of which was the discovery of three new sporadic simple groups. There X was the conjugacy class of involutions which are 3-transpositions.

• With Chris Bates, Dave Bundy and Peter Rowley, I analysed the commuting involution graph C(G,X) where X is an involution conjugacy class of G, for various choices of G. We have considered the symmetric group here, other finite Coxeter groups here, linear groups here and sporadic simple groups here and here. We have also investigated commuting graphs for arbitrary conjugacy classes in symmetric groups here.
• I extended these results to look at affine Weyl groups. It turns out that even in affine groups of type B, the diameters of commuting involution graphs can be arbitrarily large. However, restricting to the case of an affine Weyl group of type A, there is a bound on the diameter of any connected commuting involution graph [article]. My PhD student Amal Clarke and I are working on the case of type C, and it appears that here the diameter of a connected commuting involution graph is bounded above by the rank of the group.

TPP Triples: Only one paper on these so I won't go into details, but basically these are subsets of groups whose properties make them relevant to applications in fast matrix multiplication - just google Cohn and Umans and you will see what I mean.

Root Differences: Again only one paper on these [link] .

Permutations of k-sets: Suppose G is a permutation group acting on a set Omega of size n. The Livingstone Wagner Theorem says that the number of orbits of G acting on k-subsets of Omega is less than or equal to the number of orbits on (k+1)-subsets, as long as k < (n-1)/2. Here we investigate when equality occurs.
We say that G is k-free when the set-wise stabiliser of every k-subset is trivial. The primitive k-free permutation groups are described here.

## Contact details

Phone: +44 (0) 20 7631 6437
Email: s.hart@bbk.ac.uk
Room: 753
Office Hours:Fridays 4:30 - 5:45