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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 that1) $\sum_{i=1}^s A_i =J$ the all-one-matrix2) $A_1 = I$ 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 $a$.
