Over the past few months, I have been working on some project regarding flags in finite geometries with Jan De Beule and Klaus Metsch. A preprint on this should appear in the near future. In any case, there is plenty of material that does not quite fit into what we did, but seemed interesting nonetheless. Therefore, I’ll post it here.

First of all, we need to recall the definition of a flag in an incidence geometry and its type. There are exact definitions of ‘flag’, ‘incidence geometry’, and ‘type’, but since we consider rather natural geometries, I will not bother to state them. Instead, it suffices to know that a flag is a set of elements in the incidence geometry that are pairwise incident. An example illuminates this best.

**Example. **Consider the the set , then a natural incidence geometry on this ground set has as elements all subsets of and containment as the incidence relation. The type of an element is the size of the subset. Then the set is a flag of type in this geometry, just as is one of type .

The easiest flags to handle are those of of size one and a lot is known when we are dealing with flags of size one. In the previous example these are just the subsets of . We could investigate the flags of type , which boils down to the Johnson scheme , where all possible relations between two flags of type are . A symmetric and commutative algebra appears when we consider the vector space spanned by the adjacency matrices of these relations, which is called an association scheme.

In our case, we are interested in subspaces of . When considering only -dimensional subspaces, the natural relations depend on the dimension of the intersection of two such spaces. The algebra one derives from this is again symmetric and commutative is often called the -Johnson scheme , as it is a -analog. Again, when we want to consider flags of subspaces, things quickly get more complicated.

In general, the symmetry of the algebras and is easy to see, since the relations on flags of size one themselves are symmetric. Since an algebra of symmetric matrices is commutative, the latter also follows. When we consider larger flags however, symmetry is immediately gone. One can wonder what remains of the commutativity.

In the two cases we mention, this problem is in fact solved (in disguise) by Godsil and Meagher. When we consider not just flags in vector spaces, but also flags in polar and parapolar spaces and other geometries, building theory pops up, and this has been handled by Abramenko, Parkinson and Van Maldeghem, with different methods.

Buildings, or what I understand of it, are a very natural framework to investigate flags in classical incidence geometries. Although we restrict ourselves to spherical irreducible buildings, this class is already rich enough to obtain the results we are interested in. We will focus on the commutativity of the association scheme on flags of subsets of . The remarkable strength of building theory is that it allows the exact same result to be transferred to flags of subspaces in !

The result for flags of subsets of states that the association scheme (or adjacency algebra) is commutative only for flags of size one, hence only for the Johnson scheme , for some . In the article by Godsil and Meagher, this is proven using some character theory. We will sketch a purely combinatorial proof here, based on the combinatorics of compositions and partitions.

Consider the symmetric group generated by the adjacent transpositions . It can be checked that the relations between the generators are , for and if . Moreover, it turns out that this is presentation for . We can now reconstruct the association scheme as the association scheme with relations the orbitals (orbits on pairs) of on the -subsets of . In fact, there is quite some redundancy there, as we do not need the full group , but can restrict ourselves to the stabilizer of a single -subset. One can easily see that for the -subset , the stabilizer is the subgroup generated by . Now this subgroup is a Young subgroup .

This construction holds for the association scheme of flags of any type. For each flag there is a corresponding Young subgroup, and every Young subgroup corresponds to a composition of (a set of non-negative integers summing to ). In the example above, this is the composition . The commutativity of the association scheme is encoded in the induction of the trivial character of the Young subgroup to . It is commutative if and only if this character is multiplicity-free, i.e. every irreducible character of appears with multiplicity 0 or 1 in the decomposition. Here comes the kicker: the irreducible representation of are parametrized by partitions (compositions with the parts ordered in non-decreasing way) of , and the multiplicities of can be computed in a purely combinatorial way depending on these two compositions!

These coefficients are called Kostka numbers, and for our purpose it suffices to know whether they are larger than 1 or not. This becomes rather easy if you know how to compute them combinatorially: given a partition of , i.e. ordered such that , you can picture its Young tableau, a collection of left-aligned boxes with in the first row, in the second row and so on. Now given another composition you want to fill in the boxes of Young tableau above with times the number 1, times the number 2 and so on, in such a way that the rows are non-decreasing and the columns strictly increasing. The number of ways you can do this is the Kostka number .

Since , where is the irreducible character corresponding to the partition and is a Young subgroup corresponding to the composition , we need to find out for which there exists such that . When has only two parts, it is easy to see that this can never occur. However, when has at least three parts, it is a nice exercise to see that there is always at least one composition such that .

The conclusion is that for flags of subsets or subspaces, we never find a commutative association scheme, except for flags of size one. This means that a lot of tools of commutative association schemes will not work anymore for more general flags other techniques are needed to overcome this difficulty.