Association scheme on injective functions

2010-07-22T20:15:27Z

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?

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?

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.

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 $g' = \pi \circ f' \circ \tau$.

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 $a$.

Answer by Chris Godsil for Association scheme on injective functions

2010-07-28T12:49:31Z

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 :-(

Association scheme on injective functions - MathOverflow

Association scheme on injective functions

Answer by Chris Godsil for Association scheme on injective functions