This problem arises while studying the complexity of algorithms and I am quite unfamiliar with the subject.


Consider the set F of injective functions from {1..N} to {1..M}


we can define an association scheme on F x F by
(f,f') and (g,g') are in the same class if there is a permutation $\pi\in S\_M$ and a permutation $\tau \in S\_N$ such that $g = \pi \circ f \circ \tau$ and $g' = \pi \circ f' \circ \tau$.


I checked that this really defines an association scheme. In a way it is an "ordered" version of the Johnson scheme. It seems to me that it is a natural extension of the Johnson scheme, but I did not find any reference about it.



> 
> Q1: Has this association scheme ever been studied? What is its name?
> 
> 
> Q2: Can this scheme be obtained by a combination (tensor product? suprema?) of the Johnson scheme and another quantity?
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> 
> 


More precisely, I am interested in the "Bose-Mesner Algebra" point of view on this scheme. It is known that all the matrices in the algebra defined by this association scheme diagonalize in the same basis.



> 
> Q3: How can we construct/characterize these eigenspaces? 
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> 
> 


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Some background on Association Schemes.


An association scheme is a set of symmetric boolean matrices $A\_1, \dots , A\_S$ such that
1) $\sum\_{i=1}^s A\_i =J$ the all-one-matrix
2) $A\_1 = I$ the identity matrix 
3) $\forall i,j \; A\_iA\_j \in {\rm span} ( A\_i )$


The matrices $A\_i$ can be seen as adjency matrix for some graph (but I don't think it might help here)


The span{$A\_i$} defines an algebra called the Bose-Mesner Algebra. Condition (3) implies that all matrices commute so they diagonalize in the same basis.


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In the case I'm considering here, the dimension of the $A\_i$ is ${M \choose N}N!\times {M \choose N}N!$. The $A\_i$ are not explicitly defined but we know that $[A\_i]\_{fg}=[A\_i]\_{f'g'}$ if there is a permutation $\pi\in S\_M$ and a permutation $\tau \in S\_N$ such that $g = \pi \circ f \circ \tau$ and $g' = \pi \circ f' \circ \tau$.


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About the Johnson scheme: The $A\_i$ have size ${M \choose N}$. The rows and the columns of the matrices are labeled by subsets of size $N$ of {$1,\dots,M$}. (in my case, the labels are injective functions, ie. ordered sets of subsets of size $N$ of {$1,\dots,M$}.


$[A\_i]\_{ab}=[A\_i]\_{a'b'}$ if there is a permutation $\pi\in S\_M$ such that $\pi(a) = a'$ and $\pi(b) = b'$. (where $\pi(a)$ denotes the subset of {$1,\dots,M$} obtained by applying the permutation $\pi$ to the elements of the sets $a$.



Association scheme on injective functions
I don't think this scheme has a particular name, and am not aware of any study of it. Its Bose-Mesner algebra is the commutant of a multiplicity-free representation of the wreath product of $S\_m$ by $S\_n$. To get the eigenspaces you need to find the decomposition of the representation into irreducibles.


The most useful reference I know is "Harmonic Analysis on Finite Groups: Representation Theory, Gelfand Pairs and Markov Chains" by Ceccherini-Silberstein, Scarabotti and Tolli.


for what it's worth, I think there's a chance that getting the decomposition is actually
doable, but it would not be quick :-(


