association schemes, infinite schemes, semi-schemes, quasi-schemes

Question

Some questions about possibly nonsensical ideas:

1) Can you come up with a definition of an infinite association scheme ?

2) Would infinite association schemes relate to infinite groups the way association schemes relate to finite groups ? (see http://www.ams.org/bull/2006-43-02/S0273-0979-05-01077-3/S0273-0979-05-01077-3.pdf)

3) Can you define objects that relate to semigroups and quasigroups as schemes relate to groups ?

4) What combinatorial, statistical or other properties would these infinite schemes, semi-schemes and quasi-schemes have ?

Answer 1

Look for work by Zieschang, in particular his "Theory of Association Schemes", published by Springer. (There is lot of work by a lot of people, but this will get you started.)

The association schemes associated to finite groups (conjugacy class schemes) provide a significant but unrepresentative class of examples. It is not clear that viewing association schemes as generalizations of groups is particularly useful. Applications of infinite schemes to combinatorics seem to be quite scarce.

Answer 2

In "Algebraic combinatorics I: association schemes" By E. Bannai, T. Itō the theory of association schemes is described as "group theory without groups", but this is misleading as there are many association schemes that are not permutation group schemes. In a series of papers "Characters of finite quasigroups I-VII" by J.D.H. Smith and K.W. Johnson the combinatorial character theory of finite quasigroups and the theory of association schemes on which it depends is developed, and an interesting notion of a superscheme is introduced. A superscheme is a higher-dimensional analogue of an association scheme and its binary part is an association scheme (called its associated scheme :-)). In his article "Association schemes, superschemes and relations invariant under permutation groups" J.D.H. Smith says that superschemes put the groups back into the group theory without groups. In fact he shows that permutation group schemes are precisely the association schemes that are the associated scheme of a superscheme. Now, a good reference if you find this interesting is the 2007 book by Smith "An introduction to quasigroups and their representations".

Answer 3

A (probably not very brilliant) partial answer to the first question.

The following definition of infinite association scheme semi-rings makes sense: A set $A_i,i\in\mathcal I$ (with $\mathcal I$ not necessarily finite) of infinite matrices (indexed by $\mathbb N$ or $\mathbb Z$) with coefficients in $\{0,1\}$ containing the infinite identity matrix such that $\sum_{i\in \mathcal I}A_i=J$ where $J$ denotes the infinite all $1$ matrix and $A_iA_j=\sum_{k\in\mathcal I}\gamma_{i,j}^kA_k$ with $\gamma_{i,j}^k$ in $\mathbb N\cup\lbrace\infty\rbrace$ and where the last sum is finite. We require moreover the equalities $\gamma_{i,j}^k=\gamma_{j,i}^k$. All operations are then well-defined on the semiring $\sum_{i\in\mathcal I}\lambda_i A_i$ of finite sums with coefficients in $\mathbb R_{\geq 0}\cup\{\infty\}$ and can even be extended to infinite sums (this is useful since the Hadamard product identity, $J$ is an infinite sum). Negative coefficients should be avoided.

I am not convinced of the interest of such a structure.